How can you check to see if a line is parallel to another line ?

A line is parallel to another line if their direction vectors are scalar multiples .

How can you check whether two vector equations represent the same line ?

Two vector equations represent the same line if: the direction vectors are scalar multiples , and a point on one line also lies on the other line .

True or False? If two lines in 3D are not parallel then they must intersect .

False. If two lines in 3D are not parallel then they could intersect , but they do not have to intersect; they could be skew .

What are skew lines ?

Skew lines are lines that are not parallel and do not intersect .

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 .

IBMathsDPAnalysis & Approaches (AA)HLFlashcards3. Geometry & TrigonometryVector Equations of Lines

Vector Equations of Lines (DP IB Analysis & Approaches (AA))

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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.
True or False?
The Cartesian equation of a line in 3D is of the form $a x + b y + c z = d .$
False.
The Cartesian equation of a line in 3D is not of the form $a x + b y + c z = d .$ This is an equation of a plane.
The cartesian equation of a line in 3D is in the form $\frac{x - x_{0}}{l} = \frac{y - y_{0}}{m} = \frac{z - z_{0}}{n}$ .
This is given in the formula booklet.
What is the Cartesian equation of a line in 3D?
The Cartesian equation of a line in 3D is $\frac{x - x_{0}}{l} = \frac{y - y_{0}}{m} = \frac{z - z_{0}}{n}$ .
This is given in the formula booklet.
How would you write the Cartesian equation of a line if one (or more) of the components of the direction vector is zero?
For example, if the direction vector is $(\begin{matrix} 2 \\ 0 \\ 3 \end{matrix})$ .
If one (or more) of the components of the direction vector of a line is zero then you write that the corresponding variable is equal to the relevant coordinate. You would put this as a separate equation after the other equation such as $\frac{x - x_{0}}{l} = \frac{z - z_{0}}{n}, y = y_{0}$
For example, if the direction vector is $(\begin{matrix} 2 \\ 0 \\ 3 \end{matrix})$ then the Cartesian equation would look like $\frac{x - x_{0}}{2} = \frac{z - z_{0}}{3}, y = y_{0}$ .
Given the Cartesian equation of a line, how can you find a vector equation of the line?
Given the equation $\frac{x - x_{0}}{l} = \frac{y - y_{0}}{m} = \frac{z - z_{0}}{n}$ , to find a vector equation:
- set the Cartesian equation equal to $λ$ ,
- rearrange each section of the equation to find expressions for $x$ , $y$ and $z$ in terms of $λ$ ,
- the coefficients of $λ$ form the direction vector and the constant terms form the position vector of a point on the line.
An object is moving with constant velocity, $v$ , along a straight line. The position vector of the starting point is $r_{0}$ . What is an equation for the position vector of the object after time $t$ ?
Then the position of the object at the time, t can be given by $r = r_{0} + t v$ where:
- $r_{0}$ is the position vector of the object when $t = 0$ ,
- $v$ is the vector for the constant velocity,
- $t$ is the time.
An object is moving with velocity vector $v$ . How do you find the speed of the object?
An object is moving with velocity vector $v$ . You can find the speed by finding the magnitude of the velocity vector, $|v|$ .
How can you check to see if a line is parallel to another line?
A line is parallel to another line if their direction vectors are scalar multiples.
How can you check whether two vector equations represent the same line?
Two vector equations represent the same line if:
- the direction vectors are scalar multiples,
- and a point on one line also lies on the other line.
True or False?
If two lines in 3D are not parallel then they must intersect.
False.
If two lines in 3D are not parallel then they could intersect, but they do not have to intersect; they could be skew.
What are skew lines?
Skew lines are lines that are not parallel and do not intersect.
How do you determine whether two non-parallel lines intersect or are skew?
To determine whether two non-parallel lines intersect or are skew, you:
- set the vector equations equal to each other, $a + λ b = c + μ d$
- use the components to form three equations in terms of $λ$ and $μ$ ,
- solve any two of these equations simultaneously,
- check whether these values for $λ$ and $μ$ satisfy the third equation.
If the third equation is satisfied then the lines intersect, otherwise they are skew.
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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