How do you find the angles between two lines ?

The angles between two lines can be found by finding the angles between a direction vector of one line and a direction vector of the other line. You can use the scalar product to do this.

How do you find the shortest distance between two parallel lines ?

To find the shortest distance between two parallel lines : pick a point on one of the lines, find the shortest distance between this point and the other line .

IBMathsDPApplications & Interpretation (AI)HLFlashcards3. Geometry & TrigonometryVector Equations of Lines

Vector Equations of Lines (DP IB Applications & Interpretation (AI))

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FrontVector Equations of Lines
A vector equation of a line is $r = a + λ b$ .
What does the vector $a$ represent?

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A vector equation of a line is $r = a + λ b$ .
What does the vector $a$ represent?
A vector equation of a line is $r = a + λ b$ .
The vector $a$ represents a position vector of any point on the line.
A vector equation of a line is $r = a + λ b$ .
What does the vector $b$ represent?
A vector equation of a line is $r = a + λ b$ .
The vector $b$ represents a direction vector.
True or False?
A vector equation of a line is $r = a + λ b$ .
The value of $λ$ determines where a point lies on the line.
True.
A vector equation of a line is $r = a + λ b$ .
The value of $λ$ determines where a point lies on the line.
$λ$ is similar to $x$ in the equation of a line in 2D, $y = m x + c$ .
True or False?
For any line, there are an infinite number of ways of writing its equation in vector form.
True.
For any line, there are an infinite number of ways of writing its equation in vector form.
This is because there are an infinite number of choices for the vectors $a$ and $b$ .
True or False?
A line passes through the points with position vectors $a$ and $b$ . A possible equation of the line is $r = b - a + λ b$ .
False.
A line passes through the points with position vectors $a$ and $b$ . A possible equation of the line is $r = a + λ (b - a)$ .
Other options are $r = a + λ (a - b)$ , $r = b + λ (b - a)$ and $r = b + λ (a - b)$ .
If the equation of a line is given in vector form, how can you determine whether a given point lies on the line?
If the equation of a line is given in vector form, you can determine whether a given point lies on the line by:
- setting the equation of the line equal to the position vector of the point, $a + λ b = p$ ,
- forming three equations using the components of the vectors, e.g. $a_{1} + λ b_{1} = p_{1}$ ,
- solving any of the equations to find value for $λ$ ,
- checking if this value satisfies the other two equations.
If the value satisfies the other two equations then the point lies on the line.
The direction vector of a line is $(\begin{matrix} l \\ m \\ n \end{matrix})$ and the position vector of a point that lies on the line is $(\begin{matrix} x_{0} \\ y_{0} \\ z_{0} \end{matrix})$ . What is the equation of the line in parametric form?
The direction vector of a line is $(\begin{matrix} l \\ m \\ n \end{matrix})$ and the position vector of a point that lies on the line is $(\begin{matrix} x_{0} \\ y_{0} \\ z_{0} \end{matrix})$ . The equation of the line in parametric form is
$\begin{array}{rcl} x & = & x_{0} + λ l \\ y & = & y_{0} + λ m \\ z & = & z_{0} + λ n \end{array}$
This is given in the formula booklet.
How do you find the angles between two lines?
The angles between two lines can be found by finding the angles between a direction vector of one line and a direction vector of the other line. You can use the scalar product to do this.
True or False?
One angle between a line ( $r = a_{1} + λ b_{1}$ ) and another line ( $r = a_{2} + μ b_{2}$ ) is given by the equation $\cos θ = \frac{b_{1} \cdot b_{2}}{|b_{1}| |b_{2}|}$ .
True.
One angle between a line ( $r = a_{1} + λ b_{1}$ ) and another line ( $r = a_{2} + μ b_{2}$ ) is given by the equation $\cos θ = \frac{b_{1} \cdot b_{2}}{|b_{1}| |b_{2}|}$ .
How do you find the shortest distance between a point, P, and a line, $r = a + λ b$ ?
To find the shortest distance between a point, P, and a line, $r = a + λ b$ , you
- let F be the point on the line closest to P, $f = a + λ_{0} b$ ,
- find a displacement vector equation from F to P,
- form an equation by making the scalar product of this displacement vector and direction vector of the line equal to zero,
- find the value of $λ_{0}$ by solving the equation,
- substitute the value of $λ_{0}$ into the displacement vector and find the magnitude.
True or False?
If a line passes through the point A and has direction vector $b$ , then the shortest distance from the line to the point P is $\frac{|\vec{A P} \times b|}{|b|}$ .
True.
If a line passes through the point A and has direction vector $b$ , then the shortest distance from the line to the point P is $\frac{|\vec{A P} \times b|}{|b|}$ .
This formula is not given in the formula booklet.
How do you find the shortest distance between two parallel lines?
To find the shortest distance between two parallel lines:
- pick a point on one of the lines,
- find the shortest distance between this point and the other line.
How do you find the shortest distance between to skew lines $r = a_{1} + λ b_{1}$ and $r = a_{2} + μ b_{2}$ ?
To find the shortest distance between to skew lines $r = a_{1} + λ b_{1}$ and $r = a_{2} + μ b_{2}$ , you:
- find an expression for the displacement between a general point on each line, $(a_{1} + λ b_{1}) - (a_{2} + μ b_{2})$ ,
- find the vector product of the two direction vectors, $b_{1} \times b_{2}$ ,
- multiply the vector product by a constant $k$ and set it equal to the displacement vector,
- solve the simultaneous equations to find the value of $k$ ,
- the shortest distance is $|k| |b_{1} \times b_{2}|$ .

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