How can you determine if a line fully lies in a plane ?

A line fully lies in a plane if: the line and the plane are parallel , a point on the line is also a point on the plane (or vice versa).

How can you determine whether a line intersects a plane exactly once ?

A line intersects a plane exactly once if they are not parallel .

True or False? Three planes , none of which are parallel, must intersect at exactly one point .

False. Three planes , none of which are parallel, could intersect at exactly one point . However, they could also intersect along a line of intersection or there could be no mutual points of intersection for all three planes.

If there are three planes , none of which are parallel, how can you determine how, or if, they intersect ?

For three non-parallel planes, you can determine how they intersect by trying to solve the three Cartesian equations simultaneously . Alternatively, you can find the line of intersection between two of the planes and see how (or if) that intersects the third plane .

How do you find the angles between two planes ?

The angles between two planes can be found by finding the angles between a normal vector to one plane and a normal vector to the other plane.

How do you find the shortest distance between a line and a parallel plane ?

To find the shortest distance between a line and a parallel plane , you find a point on the line, find the shortest distance between that point and the plane.

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

To find the shortest distance between two parallel planes , you find a point on one of the planes, find the shortest distance between that point and the other plane.

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FrontVector Equations of Planes
A vector equation of a plane is $r = a + λ b + μ c$ .
What does the vector $a$ represent?

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A vector equation of a plane is $r = a + λ b + μ c$ .
What does the vector $a$ represent?
A vector equation of a plane is $r = a + λ b + μ c$ .
The vector $a$ represents a position vector of any point on the plane.
A vector equation of a plane is $r = a + λ b + μ c$ .
What do the vectors $b$ and $c$ represent?
A vector equation of a plane is $r = a + λ b + μ c$ .
The vectors $b$ and $c$ represent any two vectors parallel to the plane but not parallel to each other.
True or False?
For any plane, there are an infinite number of ways of writing its equation in vector form.
True.
For any plane, 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 $a$ , $b$ and $c$ .
If the equation of a plane is given in vector form, how can you determine whether a given point lies on the plane?
If the equation of a plane is given in vector form, you can determine whether a given point lies on the plane by:
- setting the equation of the plane equal to the position vector of the point, $a + λ b + μ c = d$ ,
- forming three equations using the components of the vectors, e.g. $a_{1} + λ b_{1} + μ c_{1} = d_{1}$ ,
- solving any two of the equations to find values for $λ$ and $μ$ ,
- checking if those values satisfy the third equation.
If the values satisfy the third equation then the point lies on the plane.
One way of writing an equation of a plane is $r \cdot n = a \cdot n$ , what does the vector $n$ represent?
One way of writing an equation of a plane is $r \cdot n = a \cdot n$ .
The vector $n$ represents any normal vector to the plane.
True or False?
If an equation of a plane is $a x + b y + c z = d$ , then the vector $(\begin{matrix} a \\ b \\ c \end{matrix})$ is normal to the plane.
True.
If an equation of a plane is $a x + b y + c z = d$ , then the vector $(\begin{matrix} a \\ b \\ c \end{matrix})$ is normal to the plane.
If the equation of a plane is given in the form $r = a + λ b + μ c$ , how can you find a normal vector to the plane using the vectors in the equation?
If the equation of a plane is given in the form $r = a + λ b + μ c$ , you can find a normal vector to the plane by taking the vector product of $b$ with $c$ , i.e. $n = b \times c$ .
If the equation of a plane is given in the form $a x + b y + c z = d$ , how can you find the value of $d$ if you are given the position vector of a point on the plane.
If the equation of a plane is given in the form $a x + b y + c z = d$ , you can find the value of $d$ by substituting the coordinates of the given point into the equation.
True or False?
$2 x + 3 y + z = 5$ and $4 x + 6 y + 2 z = - 3$ are both equations of the same plane?
False.
$2 x + 3 y + z = 5$ and $4 x + 6 y + 2 z = - 3$ are equations of different planes.
Two Cartesian equations can only represent the same plane if the equations as a whole are scalar multiples of each other, e.g. $2 x + 3 y + z = 5$ and $4 x + 6 y + 2 z = 10$ .
How can you check to see if a line is parallel to a plane?
A line is parallel to a plane if a direction vector of the line, $b$ , is perpendicular to a normal vector to the plane, $n$ . They are parallel if $b \cdot n = 0$ .
How can you determine if a line fully lies in a plane?
A line fully lies in a plane if:
- the line and the plane are parallel,
- a point on the line is also a point on the plane (or vice versa).
How can you determine whether a line intersects a plane exactly once?
A line intersects a plane exactly once if they are not parallel.
If a line ( $r = a + λ b$ ) intersects a plane ( $a x + b y + c z = d$ ) exactly once, how do you find the point of intersection?
To find the point of intersection between a line and a plane:
- use the equation of the line to write expressions for $x$ , $y$ and $z$ in terms of $λ$ ,
- substitute the expressions into the equation of the plane,
- solve for $λ$ ,
- substitute the value of $λ$ into the expressions for $x$ , $y$ and $z$ to get the coordinates.
How can you check to see if two planes are parallel?
Two planes are parallel if a normal vector to one of them is a scalar multiple of a normal vector to the other.
A normal vector can be identified from anequation of the plane by looking at $n$ , $b \times c$ or $(\begin{matrix} a \\ b \\ c \end{matrix})$ .
If two planes are not parallel, how do you find the equation of the line of intersection?
To find the line of intersection between two planes:
- write both equations in Cartesian form,
- choose one variable and call it $λ$ ,
- solve the equations simultaneously to find expressions for the other two variables in terms of $λ$ ,
- use the expressions for $x$ , $y$ and $z$ to form a vector equation of the line.
True or False?
Three planes, none of which are parallel, must intersect at exactly one point.
False.
Three planes, none of which are parallel, could intersect at exactly one point. However, they could also intersect along a line of intersection or there could be no mutual points of intersection for all three planes.
If there are three planes, none of which are parallel, how can you determine how, or if, they intersect?
For three non-parallel planes, you can determine how they intersect by trying to solve the three Cartesian equations simultaneously.
Alternatively, you can find the line of intersection between two of the planes and see how (or if) that intersects the third plane.
How can you find the acute angle between a line ( $r = a + λ b$ ) and a plane ( $r \cdot n = a \cdot n$ )?
To find the acute angle between a line ( $r = a + λ b$ ) and a plane ( $r \cdot n = a \cdot n$ ), you find the acute angle between the direction vector of the line, $b$ , and the normal vector to the plane, $n$ , and then subtract this angle from 90°.
How do you find the angles between two planes?
The angles between two planes can be found by finding the angles between a normal vector to one plane and a normal vector to the other plane.
True or False?
One angle between a line ( $r = a + λ b$ ) and a plane ( $r \cdot n = a \cdot n$ ) is given by the equation $\cos α = \frac{|b \cdot n|}{|b| |n|}$ .
False.
One angle between a line ( $r = a + λ b$ ) and a plane ( $r \cdot n = a \cdot n$ ) is found by finding the acute angle that satisfies the equation $\cos α = \frac{|b \cdot n|}{|b| |n|}$ and then subtracting it from 90°.
True or False?
One angle between a plane ( $r \cdot n_{1} = a_{1} \cdot n_{1}$ ) and another plane ( $r \cdot n_{2} = a_{2} \cdot n_{2}$ ) is given by the equation $\cos θ = \frac{n_{1} \cdot n_{2}}{|n_{1}| |n_{2}|}$ .
True.
One angle between a plane ( $r \cdot n_{1} = a_{1} \cdot n_{1}$ ) and another plane ( $r \cdot n_{2} = a_{2} \cdot n_{2}$ ) is given by the equation $\cos θ = \frac{n_{1} \cdot n_{2}}{|n_{1}| |n_{2}|}$ .
How do you find the shortest distance between a point, P, and a plane, $a x + b y + c z = d$ ?
To find the shortest distance between a point, P, and a plane, $a x + b y + c z = d$ , you
- find a normal vector to the plane, $n = (\begin{matrix} a \\ b \\ c \end{matrix})$ ,
- find a vector equation of the line that is perpendicular to the plane and passes through the point, $r = p + λ n$ ,
- find the value of $λ$ at the point of intersection between the line and the plane,
- find the shortest distance by calculating $|λ n|$ .
How do you find the shortest distance between a line and a parallel plane?
To find the shortest distance between a line and a parallel plane, you
- find a point on the line,
- find the shortest distance between that point and the plane.
How do you find the shortest distance between two parallel planes?
To find the shortest distance between two parallel planes, you
- find a point on one of the planes,
- find the shortest distance between that point and the other plane.

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