Vector Properties (DP IB Analysis & Approaches (AA))

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  • What is a scalar?

    A scalar is a quantity without direction.

  • True or False?

    Scalars can never be negative.

    False.

    Scalars can be negative.

    Most real-life scalars, such as speed or distance, can never be negative.

  • What is a vector?

    A vector is a quantity which has a direction.

  • Write the vector open parentheses table row 2 row 3 row 1 end table close parentheses in terms of the base vectors bold italic i, bold italic j and bold italic k.

    open parentheses table row 2 row 3 row 1 end table close parentheses equals 2 bold italic i plus 3 bold italic j plus bold italic k.

  • Which vector is represented in the diagram below: AB with rightwards arrow on top or BA with rightwards arrow on top?

    A line segment labelled A and B with an arrow pointing from A to B.

    The arrow points from B to A. Therefore the vector is BA with rightwards arrow on top.

  • How do you know if two vectors are parallel?

    Two vectors are parallel if they are scalar multiples of each other. This means you can multiply one of them by a scalar to get the other one.

    For example, open parentheses table row 2 row cell negative 3 end cell row 5 end table close parentheses is parallel to open parentheses table row 12 row cell negative 18 end cell row 30 end table close parentheses but is not parallel to open parentheses table row 4 row cell negative 6 end cell row cell negative 10 end cell end table close parentheses.

  • What is the geometric relationship between the vectors bold italic v and negative bold italic v?

    bold italic v and negative bold italic v both have the same length but they are in opposite directions.

  • How do you multiply a vector by a scalar?

    For example, if bold italic v equals open parentheses table row 2 row cell negative 3 end cell row 4 end table close parentheses then what is 5 bold italic v?

    To multiply a vector by a scalar, you multiply each component by the scalar.

    For example, if bold italic v equals open parentheses table row 2 row cell negative 3 end cell row 4 end table close parentheses then 5 bold italic v equals open parentheses table row cell 5 cross times 2 end cell row cell 5 cross times open parentheses negative 3 close parentheses end cell row cell 5 cross times 4 end cell end table close parentheses.

  • How do you add or subtract two vectors?

    For example, what is open parentheses table row 2 row 1 row 5 end table close parentheses plus open parentheses table row 3 row 0 row cell negative 7 end cell end table close parentheses?

    To add or subtract two vectors, you add or subtract the corresponding components together.

    For example, open parentheses table row 2 row 1 row 5 end table close parentheses plus open parentheses table row 3 row 0 row cell negative 7 end cell end table close parentheses equals open parentheses table row cell 2 plus 3 end cell row cell 1 plus 0 end cell row cell 5 plus open parentheses negative 7 close parentheses end cell end table close parentheses.

  • What is the geometric relationship between bold italic a, bold italic b and bold italic a plus bold italic b?

    The vector bold italic a plus bold italic b can be drawn by:

    • drawing bold italic a,

    • drawing bold italic b starting at the end of bold italic a,

    • joining the start of bold italic a to the end of bold italic b.

    Triangle with black lines representing vectors 'a' and 'b' that combine to form a green line labeled 'a + b' at the bottom, showing vector addition.
  • True or False?

    If you draw two vectors bold italic t and bold italic u so that they both end at the same point, then the vector bold italic t minus bold italic u is the vector that goes from the start of bold italic t to the end of bold italic u.

    Diagram with vectors labelled B, C, D. Vector t points to B from vertex C. Vector u points to D from vertex C. Both vectors form an angle at C.

    True.

    If you draw two vectors bold italic t and bold italic u so that they both end at the same point, then the vector bold italic t minus bold italic u is the vector that goes from the start of bold italic t to the end of bold italic u.

    A triangle with vertices labelled B, C, and D. Sides BC and CD are marked with vectors t and u respectively. The side BD is marked with t-u.
  • What is meant by a position vector of a point?

    A position vector of a point is the vector starting from the origin and ending at the point.

    For example, the position vector of the point open parentheses 3 comma space 1 comma space 4 close parentheses is open parentheses table row 3 row 1 row 4 end table close parentheses.

  • What is meant by a displacement vector such as AB with rightwards arrow on top?

    A displacement vector is the vector starting at one point and ending at another point.

    For example, AB with rightwards arrow on top is the vector from straight A to straight B.

  • True or False?

    For any three points straight A, straight B and straight C, AB with rightwards arrow on top equals AC with rightwards arrow on top plus CB with rightwards arrow on top.

    True.

    For any three points straight A, straight B and straight C, AB with rightwards arrow on top equals AC with rightwards arrow on top plus CB with rightwards arrow on top.

  • Write the vector BA with rightwards arrow on top in terms of AB with rightwards arrow on top.

    BA with rightwards arrow on top equals negative AB with rightwards arrow on top.

  • True or False?

    AB with rightwards arrow on top equals stack OA blank with rightwards arrow on top minus OB with rightwards arrow on top.

    False.

    AB with rightwards arrow on top equals stack OB blank with rightwards arrow on top minus OA with rightwards arrow on top. Start with the position vector of the endpoint.

  • If bold italic v is a vector, what is represented by the notation open vertical bar bold italic v close vertical bar?

    open vertical bar bold italic v close vertical bar represents the magnitude (or length) of the vector bold italic v.

  • How do you calculate the magnitude of a vector ?

    To calculate the magnitude of a vector:

    • square each component,

    • add them together,

    • take the positive square root.

    For example, open vertical bar open parentheses table row cell v subscript 1 end cell row cell v subscript 2 end cell row cell v subscript 3 end cell end table close parentheses close vertical bar equals square root of v subscript 1 squared plus v subscript 2 squared plus v subscript 3 squared end root. This is given in the formula booklet.

  • True or False?

    open vertical bar open parentheses table row 3 row cell negative 2 end cell row 4 end table close parentheses close vertical bar equals square root of 3 squared minus 2 squared plus 4 squared end root.

    False.

    open vertical bar open parentheses table row 3 row cell negative 2 end cell row 4 end table close parentheses close vertical bar not equal to square root of 3 squared minus 2 squared plus 4 squared end root. open vertical bar open parentheses table row 3 row cell negative 2 end cell row 4 end table close parentheses close vertical bar equals square root of 3 squared plus open parentheses negative 2 close parentheses squared plus 4 squared end root.

    You can ignore the negative sign of any components because when they are squared they become positive.

  • How can use vectors to find the distance between two points?

    To find the distance between two points,

    • find the displacement vector between then, AB with rightwards arrow on top equals OB with rightwards arrow on top minus OA with rightwards arrow on top,

    • find the magnitude of this vector, open vertical bar AB with rightwards arrow on top close vertical bar.

  • What is a unit vector?

    A unit vector is a vector with a magnitude of 1.

  • How can you find a unit vector in the same direction as the vector bold italic v?

    To find a unit vector in the same direction as the vector bold italic v, you can divide the components of bold italic v by the magnitude of bold italic v, i.e. fraction numerator bold italic v over denominator open vertical bar bold italic v close vertical bar end fraction.

  • What is represented by the notation bold italic a times bold italic b where bold italic a and bold italic b are vectors?

    The notation bold italic a times bold italic b represents the scalar product of the vectors bold italic a and bold italic b.

  • True or False?

    If the scalar product of two vectors is positive then there is an acute angle between them.

    True.

    If the scalar product of two vectors is positive then there is an acute angle between them.

  • True or False?

    If the scalar product of two vectors is negative then there is an acute angle between them.

    False.

    If the scalar product of two vectors is negative then there is an obtuse angle between them.

  • How do you calculate the scalar product bold italic a times bold italic b?

    To calculate bold italic a times bold italic b, you multiply each component of bold italic a by the corresponding components of bold italic b and then add the products together.

    open parentheses table row cell v subscript 1 end cell row cell v subscript 2 end cell row cell v subscript 3 end cell end table close parentheses times open parentheses table row cell w subscript 1 end cell row cell w subscript 2 end cell row cell w subscript 3 end cell end table close parentheses equals v subscript 1 w subscript 1 plus v subscript 2 w subscript 2 plus v subscript 3 w subscript 3. This is given in the formula booklet.

  • True or False?

    The order of a scalar product does not matter, i.e. bold italic v times bold italic w equals bold italic w times bold italic v.

    True.

    The order of a scalar product does not matter, i.e. bold italic v times bold italic w equals bold italic w times bold italic v.

  • True or False?

    bold italic u times open parentheses bold italic v blank plus bold italic w close parentheses equals blank bold italic u times bold italic v plus blank bold italic u times bold italic w.

    True.

    bold italic u times open parentheses bold italic v blank plus bold italic w close parentheses equals blank bold italic u times bold italic v plus blank bold italic u times bold italic w.

  • True or False?

    k open parentheses bold italic v times bold italic w close parentheses equals open parentheses k bold italic v close parentheses times open parentheses k bold italic w close parentheses, where k is a scalar.

    False.

    k open parentheses bold italic v times bold italic w close parentheses equals open parentheses k bold italic v close parentheses times bold italic w equals bold italic v times open parentheses k bold italic w close parentheses, where k is a scalar.

    If you multiply a scalar product by a scalar then this is the same as multiplying one of the vectors by the scalar before taking the scalar product. This works just like normal numbers.

  • For any vector bold italic v, what is bold italic v times bold italic v equal to?

    For any vector bold italic v, bold italic v times bold italic v equals open vertical bar bold italic v close vertical bar squared.

  • What is the formula that connects the scalar product of two vectors (bold italic v and bold italic w) and an angle between the two vectors (theta)?

    The formula that connects the scalar product of two vectors (bold italic v and bold italic w) and an angle between the two vectors (theta) is bold italic v times bold italic w equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar cos space theta.

    This is given in the formula booklet as cos space theta equals fraction numerator v subscript 1 w subscript 1 plus v subscript 2 w subscript 2 plus v subscript 3 w subscript 3 over denominator open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar end fraction.

  • If bold italic v times bold italic w equals 0, where bold italic v and bold italic w are non-zero vectors, what is the geometrical relationship between the vectors?

    If bold italic v times bold italic w equals 0, where bold italic v and bold italic w are non-zero vectors, then the vectors are perpendicular.

  • True or False?

    If bold italic v and bold italic w are parallel vectors then open vertical bar bold italic v times bold italic w close vertical bar equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar.

    True.

    If bold italic v and bold italic w are parallel vectors then open vertical bar bold italic v times bold italic w close vertical bar equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar.

  • What is meant by the angle between two vectors?

    The angle between two vectors is the angle that is between the two vectors when they are both drawn from the same starting point.

    Two vectors starting at the same point with the angle between them.
  • True or False?

    If bold italic v and bold italic w are two non-zero perpendicular vectors, then bold italic v times bold italic w equals 1.

    False.

    If bold italic v and bold italic w are two non-zero perpendicular vectors, then bold italic v times bold italic w equals 0.

  • What is represented by the notation bold italic a cross times bold italic b where bold italic a and bold italic b are vectors?

    The notation bold italic a cross times bold italic b represents the vector (cross) product of the vectors bold italic a and bold italic b.

  • Explain what the vector product of bold italic v and bold italic w represents geometrically.

    bold italic v cross times bold italic w is a vector that is perpendicular to both the vectors bold italic v and bold italic w.

  • True or False?

    If begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator v subscript 1 over denominator table row cell v subscript 2 end cell row cell v subscript 3 end cell end table end fraction close parentheses end style and  begin mathsize 16px style bold italic w equals blank open parentheses fraction numerator w subscript 1 over denominator table row cell w subscript 2 end cell row cell w subscript 3 end cell end table end fraction close parentheses end style, then bold italic v cross times bold italic w equals blank open parentheses table row cell v subscript 2 w subscript 3 minus blank v subscript 3 w subscript 2 end cell row cell v subscript 1 w subscript 3 minus blank v subscript 3 w subscript 1 end cell row cell v subscript 1 w subscript 2 minus blank v subscript 2 w subscript 1 end cell end table close parentheses.

    False.

    If begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator v subscript 1 over denominator table row cell v subscript 2 end cell row cell v subscript 3 end cell end table end fraction close parentheses end style and  begin mathsize 16px style bold italic w equals blank open parentheses fraction numerator w subscript 1 over denominator table row cell w subscript 2 end cell row cell w subscript 3 end cell end table end fraction close parentheses end style, then bold italic v cross times bold italic w equals blank open parentheses table row cell v subscript 2 w subscript 3 minus blank v subscript 3 w subscript 2 end cell row cell v subscript 3 w subscript 1 minus blank v subscript 1 w subscript 3 end cell row cell v subscript 1 w subscript 2 minus blank v subscript 2 w subscript 1 end cell end table close parentheses. This is given in the formula booklet.

    The pattern for each component is v subscript n e x t end subscript w subscript p r e v i o u s end subscript minus v subscript p r e v i o u s end subscript w subscript n e x t end subscript where the numbers are in the sequence 1, 2, 3, 1.

    For example, the second component is v subscript 3 w subscript 1 minus blank v subscript 1 w subscript 3.

  • True or False?

    The order of a vector product does not matter, i.e. bold italic v cross times bold italic w equals bold italic w cross times bold italic v.

    False.

    The order of a vector product does matter, if the vectors are flipped then the sign of the vector product changes i.e. bold italic v cross times bold italic w equals negative bold italic w cross times bold italic v.

  • True or False?

    bold italic u cross times open parentheses bold italic v blank plus bold italic w close parentheses equals blank bold italic u cross times bold italic v plus blank bold italic u cross times bold italic w.

    True.

    bold italic u cross times open parentheses bold italic v blank plus bold italic w close parentheses equals blank bold italic u cross times bold italic v plus blank bold italic u cross times bold italic w.

  • For any vector v, what is bold italic v cross times bold italic v equal to?

    For any vector v, bold italic v cross times bold italic v equals bold 0 where bold 0 is the zero vector.

  • True or False?

    If two vectors are parallel then their vector product is equal to the zero vector.

    True.

    If two vectors are parallel then their vector product is equal to the zero vector.

  • True or False?

    If bold italic v and bold italic w are perpendicular vectors then open vertical bar bold italic v cross times bold italic w close vertical bar equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar.

    True.

    If bold italic v and bold italic w are perpendicular vectors then open vertical bar bold italic v cross times bold italic w close vertical bar equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar.

  • If bold italic v cross times bold italic w equals bold 0, where bold italic v and bold italic w are non-zero vectors, what is the geometrical relationship between the vectors?

    If bold italic v cross times bold italic w equals bold 0, where bold italic v and bold italic w are non-zero vectors, then the vectors are parallel.

  • What is the formula that connects the vector product of two vectors (bold italic v and bold italic w) and an angle between the two vectors (theta)?

    The formula that connects the vector product of two vectors (bold italic v and bold italic w) and an angle between the two vectors (theta) is bold italic v cross times bold italic w equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar sin space theta space bold italic n, where bold italic n is a unit normal vector that is perpendicular to both bold italic v and bold italic w.

    This is given in the formula booklet without the normal vector by using magnitudes: open vertical bar bold italic v cross times bold italic w close vertical bar equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar sin space theta.

  • How can you use the vector product to find the area of a parallelogram?

    You can use the vector product to find the area of a parallelogram,

    • write one side as a displacement vector, bold italic v,

    • write a side that is not parallel to the first side as a displacement vector, bold italic w,

    • the area is equal to the magnitude of the vector product of these two vectors, A equals open vertical bar bold italic v cross times bold italic w close vertical bar.

    This is given in the formula booklet.

  • How can you use the vector product to find the area of a triangle?

    You can use the vector product to find the area of a triangle,

    • write one side as a displacement vector, bold italic v,

    • write another side as a displacement vector, bold italic w,

    • the area is equal to half of the magnitude of the vector product of these two vectors, A equals 1 half open vertical bar bold italic v cross times bold italic w close vertical bar.

    This is not given in the formula booklet. It uses the fact that two identical triangles can form a parallelogram.

  • How can you use vectors to prove that a shape is a parallelogram?

    To use vectors to prove that a shape is a parallelogram, you need to:

    • Show that there are two pairs of parallel sides

    • Show that the opposite sides are of equal length

      • The vectors opposite each other will be equal

  • How can you use vectors to prove that a shape is a rectangle?

    To use vectors to prove that a shape is a rectangle, you need to show that the displacement vectors for opposite sides are equal (direction and length) and that displacement vectors for adjacent sides are perpendicular (using the scalar product).

  • How can you use vectors to prove that a shape is a rhombus?

    To use vectors to prove that a shape is a rhombus, you need to show that the displacement vectors for opposite sides are equal and that displacement vectors for all the sides have the same magnitude.

  • True or False?

    If the point straight X divides the line segment AB in the ratio 2:3, then AX with rightwards arrow on top equals 2 over 3 AB with rightwards arrow on top.

    False.

    If the point straight X divides the line segment AB in the ratio 2:3, then AX with rightwards arrow on top equals fraction numerator 2 over denominator 2 plus 3 end fraction AB with rightwards arrow on top.

  • If the point straight A has position vector bold italic a and the point straight B has position vector bold italic b then write an expression for the position vector of the midpoint of the line segment AB.

    If the point straight A has position vector bold italic a and the point straight B has position vector bold italic b then the position vector of the midpoint of the line segment AB is 1 half open parentheses bold italic a plus bold italic b close parentheses.

  • If the displacement vectors AB with rightwards arrow on top and AC with rightwards arrow on top are scalar multiples, then what is the geometric relationship between the points straight A, straight B and straight C?

    If the displacement vectors AB with rightwards arrow on top and AC with rightwards arrow on top are scalar multiples, then the points straight A, straight B and straight C line on the same straight line (they are collinear).