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.

A vector is a quantity which has a direction .

What is a unit vector ?

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

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.

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 .

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 .

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Vector Properties (DP IB Applications & Interpretation (AI))

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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 $(\begin{matrix} 2 \\ 3 \\ 1 \end{matrix})$ in terms of the base vectors $i$ , $j$ and $k$ .
$(\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}) = 2 i + 3 j + k$ .
Which vector is represented in the diagram below: $\vec{AB}$ or $\vec{BA}$ ?
The arrow points from B to A. Therefore the vector is $\vec{BA}$ .
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, $(\begin{matrix} 2 \\ - 3 \\ 5 \end{matrix})$ is parallel to $(\begin{matrix} 12 \\ - 18 \\ 30 \end{matrix})$ but is not parallel to $(\begin{matrix} 4 \\ - 6 \\ - 10 \end{matrix})$ .
What is the geometric relationship between the vectors $v$ and $- v$ ?
$v$ and $- v$ both have the same length but they are in opposite directions.
How do you multiply a vector by a scalar?
For example, if $v = (\begin{matrix} 2 \\ - 3 \\ 4 \end{matrix})$ then what is $5 v$ ?
To multiply a vector by a scalar, you multiply each component by the scalar.
For example, if $v = (\begin{matrix} 2 \\ - 3 \\ 4 \end{matrix})$ then $5 v = (\begin{matrix} 5 \times 2 \\ 5 \times (- 3) \\ 5 \times 4 \end{matrix})$ .
How do you add or subtract two vectors?
For example, what is $(\begin{matrix} 2 \\ 1 \\ 5 \end{matrix}) + (\begin{matrix} 3 \\ 0 \\ - 7 \end{matrix})$ ?
To add or subtract two vectors, you add or subtract the corresponding components together.
For example, $(\begin{matrix} 2 \\ 1 \\ 5 \end{matrix}) + (\begin{matrix} 3 \\ 0 \\ - 7 \end{matrix}) = (\begin{matrix} 2 + 3 \\ 1 + 0 \\ 5 + (- 7) \end{matrix})$ .
What is the geometric relationship between $a$ , $b$ and $a + b$ ?
The vector $a + b$ can be drawn by:
- drawing $a$ ,
- drawing $b$ starting at the end of $a$ ,
- joining the start of $a$ to the end of $b$ .
True or False?
If you draw two vectors $t$ and $u$ so that they both end at the same point, then the vector $t - u$ is the vector that goes from the start of $t$ to the end of $u$ .
True.
If you draw two vectors $t$ and $u$ so that they both end at the same point, then the vector $t - u$ is the vector that goes from the start of $t$ to the end of $u$ .
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 $(3, 1, 4)$ is $(\begin{matrix} 3 \\ 1 \\ 4 \end{matrix})$ .
What is meant by a displacement vector such as $\vec{AB}$ ?
A displacement vector is the vector starting at one point and ending at another point.
For example, $\vec{AB}$ is the vector from $A$ to $B$ .
True or False?
For any three points $A$ , $B$ and $C$ , $\vec{AB} = \vec{AC} + \vec{CB}$ .
True.
For any three points $A$ , $B$ and $C$ , $\vec{AB} = \vec{AC} + \vec{CB}$ .
Write the vector $\vec{BA}$ in terms of $\vec{AB}$ .
$\vec{BA} = - \vec{AB}$ .
True or False?
$\vec{AB} = \vec{OA} - \vec{OB}$ .
False.
$\vec{AB} = \vec{OB} - \vec{OA}$ . Start with the position vector of the endpoint.
If $v$ is a vector, what is represented by the notation $|v|$ ?
$|v|$ represents the magnitude (or length) of the vector $v$ .
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, $|(\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \end{matrix})| = \sqrt{{v_{1}}^{2} + {v_{2}}^{2} + {v_{3}}^{2}}$ . This is given in the formula booklet.
True or False?
$|(\begin{matrix} 3 \\ - 2 \\ 4 \end{matrix})| = \sqrt{3^{2} - 2^{2} + 4^{2}}$ .
False.
$|(\begin{matrix} 3 \\ - 2 \\ 4 \end{matrix})| \neq \sqrt{3^{2} - 2^{2} + 4^{2}}$ . $|(\begin{matrix} 3 \\ - 2 \\ 4 \end{matrix})| = \sqrt{3^{2} + {(- 2)}^{2} + 4^{2}}$ .
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, $\vec{AB} = \vec{OB} - \vec{OA}$ ,
- find the magnitude of this vector, $|\vec{AB}|$ .
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 $v$ ?
To find a unit vector in the same direction as the vector $v$ , you can divide the components of $v$ by the magnitude of $v$ , i.e. $\frac{v}{|v|}$ .
What is represented by the notation $a \cdot b$ where $a$ and $b$ are vectors?
The notation $a \cdot b$ represents the scalar product of the vectors $a$ and $b$ .
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 $a \cdot b$ ?
To calculate $a \cdot b$ , you multiply each component of $a$ by the corresponding components of $b$ and then add the products together.
$(\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \end{matrix}) \cdot (\begin{matrix} w_{1} \\ w_{2} \\ w_{3} \end{matrix}) = v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}$ . This is given in the formula booklet.
What is the formula that connects the scalar product of two vectors ( $v$ and $w$ ) and an angle between the two vectors ( $θ$ )?
The formula that connects the scalar product of two vectors ( $v$ and $w$ ) and an angle between the two vectors ( $θ$ ) is $v \cdot w = |v| |w| \cos θ$ .
This is given in the formula booklet as $\cos θ = \frac{v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}}{|v| |w|}$ .
If $v \cdot w = 0$ , where $v$ and $w$ are non-zero vectors, what is the geometrical relationship between the vectors?
If $v \cdot w = 0$ , where $v$ and $w$ are non-zero vectors, then the vectors are perpendicular.
True or False?
If $v$ and $w$ are parallel vectors then $|v \cdot w| = |v| |w|$ .
True.
If $v$ and $w$ are parallel vectors then $|v \cdot w| = |v| |w|$ .
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.
True or False?
If $v$ and $w$ are two non-zero perpendicular vectors, then $v \cdot w = 1$ .
False.
If $v$ and $w$ are two non-zero perpendicular vectors, then $v \cdot w = 0$ .
What is represented by the notation $a \times b$ where $a$ and $b$ are vectors?
The notation $a \times b$ represents the vector (cross) product of the vectors $a$ and $b$ .
Explain what the vector product of $v$ and $w$ represents geometrically.
$v \times w$ is a vector that is perpendicular to both the vectors $v$ and $w$ .
True or False?
If $v = (\binom{v_{1}}{\begin{matrix} v_{2} \\ v_{3} \end{matrix}})$ and $w = (\binom{w_{1}}{\begin{matrix} w_{2} \\ w_{3} \end{matrix}})$ , then $v \times w = (\begin{matrix} v_{2} w_{3} - v_{3} w_{2} \\ v_{1} w_{3} - v_{3} w_{1} \\ v_{1} w_{2} - v_{2} w_{1} \end{matrix})$ .
False.
If $v = (\binom{v_{1}}{\begin{matrix} v_{2} \\ v_{3} \end{matrix}})$ and $w = (\binom{w_{1}}{\begin{matrix} w_{2} \\ w_{3} \end{matrix}})$ , then $v \times w = (\begin{matrix} v_{2} w_{3} - v_{3} w_{2} \\ v_{3} w_{1} - v_{1} w_{3} \\ v_{1} w_{2} - v_{2} w_{1} \end{matrix})$ . This is given in the formula booklet.
The pattern for each component is $v_{n e x t} w_{p r e v i o u s} - v_{p r e v i o u s} w_{n e x t}$ where the numbers are in the sequence 1, 2, 3, 1.
For example, the second component is $v_{3} w_{1} - v_{1} w_{3}$ .
True or False?
The order of a vector product does not matter, i.e. $v \times w = w \times v$ .
False.
The order of a vector product does matter, if the vectors are flipped then the sign of the vector product changes i.e. $v \times w = - w \times v$ .
For any vector $v$ , what is $v \times v$ equal to?
For any vector $v$ , $v \times v = 0$ where $0$ 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 $v$ and $w$ are perpendicular vectors then $|v \times w| = |v| |w|$ .
True.
If $v$ and $w$ are perpendicular vectors then $|v \times w| = |v| |w|$ .
If $v \times w = 0$ , where $v$ and $w$ are non-zero vectors, what is the geometrical relationship between the vectors?
If $v \times w = 0$ , where $v$ and $w$ are non-zero vectors, then the vectors are parallel.
What is the formula that connects the vector product of two vectors ( $v$ and $w$ ) and an angle between the two vectors ( $θ$ )?
The formula that connects the vector product of two vectors ( $v$ and $w$ ) and an angle between the two vectors ( $θ$ ) is $v \times w = |v| |w| \sin θ n$ , where $n$ is a unit normal vector that is perpendicular to both $v$ and $w$ .
This is given in the formula booklet without the normal vector by using magnitudes: $|v \times w| = |v| |w| \sin θ$ .
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, $v$ ,
- write a side that is not parallel to the first side as a displacement vector, $w$ ,
- the area is equal to the magnitude of the vector product of these two vectors, $A = |v \times w|$ .
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, $v$ ,
- write another side as a displacement vector, $w$ ,
- the area is equal to half of the magnitude of the vector product of these two vectors, $A = \frac{1}{2} |v \times w|$ .
This is not given in the formula booklet. It uses the fact that two identical triangles can form a parallelogram.
How do you find the component of the vector $a$ that is acting in the direction of the vector $b$ if the angle between the vectors is unknown?
The component of the vector $a$ that is acting in the direction of the vector $b$ is equal to $\frac{a \cdot b}{| b |}$ .
This is not given in your exam formula booklet.
How do you find the component of the vector $a$ that is acting perpendicular to the vector $b$ in the plane formed by the two vectors if the angle between the vectors is unknown?
The component of the vector $a$ that is acting perpendicular to the vector $b$ in the plane formed by the two vectors is equal to $\frac{|a \times b|}{|b|}$ .
This is not given in your exam formula booklet.
True or False?
If the vector $a$ is acting at an angle $θ$ to the vector $b$ , then the component of $a$ that is acting parallel to $b$ is equal to $|a| \sin θ$ .
False.
If the vector $a$ is acting at an angle $θ$ to the vector $b$ , then the component of $a$ that is acting parallel to $b$ is equal to $|a| \cos θ$ .
If the vector $a$ is acting at an angle $θ$ to the vector $b$ , then what is the component of $a$ that is acting perpendicular to $b$ ?
If the vector $a$ is acting at an angle $θ$ to the vector $b$ , then the component of $a$ that is acting perpendicular to $b$ is equal to $|a| \sin θ$ .
What is an expression for the component of $a$ that is labelled x?
The component of $a$ that is labelled x is equal to $|a| \cos θ$ .
It is also equal to $\frac{a \cdot b}{|b|}$ .
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 $X$ divides the line segment $AB$ in the ratio 2:3, then $\vec{AX} = \frac{2}{3} \vec{AB}$ .
False.
If the point $X$ divides the line segment $AB$ in the ratio 2:3, then $\vec{AX} = \frac{2}{2 + 3} \vec{AB}$ .
If the point $A$ has position vector $a$ and the point $B$ has position vector $b$ then write an expression for the position vector of the midpoint of the line segment $AB$ .
If the point $A$ has position vector $a$ and the point $B$ has position vector $b$ then the position vector of the midpoint of the line segment $AB$ is $\frac{1}{2} (a + b)$ .
If the displacement vectors $\vec{AB}$ and $\vec{AC}$ are scalar multiples, then what is the geometric relationship between the points $A$ , $B$ and $C$ ?
If the displacement vectors $\vec{AB}$ and $\vec{AC}$ are scalar multiples, then the points $A$ , $B$ and $C$ line on the same straight line (they are collinear).

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