MCV4U · Unit 6 test review

Lines & Planes: Test Review

Everything for the Unit 6 test on one page: the key facts from all 5 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

Lines in 2-Space

Scalar equation of a line in 2-space: $A x + B y + C = 0$ , with normal $n = (A, B)$ .
Vector equation: $r = r_{0} + t m$ , $t \in R$ ; parametric: $x = x_{0} + t m_{1}$ , $y = y_{0} + t m_{2}$ .
From normal $(A, B)$ , a direction is $(B, - A)$ ; from direction $(m_{1}, m_{2})$ , a normal is $(m_{2}, - m_{1})$ .
Slope $= \frac{m _{2}}{m _{1}}$ (when $m_{1} \neq = 0$ ) $= - \frac{A}{B}$ (when $B \neq = 0$ ).
Two lines in 2-space: one point if the normals are not parallel; otherwise coincident (infinitely many) or parallel and distinct (none).

Lines in 3-Space

A single equation $A x + B y + C z + D = 0$ is a plane; a line in 3-space needs two equations (two intersecting planes).
Vector: $r = r_{0} + t m$ ; parametric: $x = x_{0} + t m_{1}$ , $y = y_{0} + t m_{2}$ , $z = z_{0} + t m_{3}$ .
Symmetric: $\frac{x - x _{0}}{m _{1}} = \frac{y - y _{0}}{m _{2}} = \frac{z - z _{0}}{m _{3}}$ (each $m_{i} \neq = 0$ ; a zero component makes that coordinate constant).
Parallel directions: coincident or parallel and distinct. Non-parallel directions: intersecting or skew; the third equation decides.
Angle between lines: $cos θ = \frac{∣ m _{1} \cdot m _{2} ∣}{∣ m _{1} ∣∣ m _{2} ∣};$ perpendicular when $m_{1} \cdot m_{2} = 0$ .

Planes: Normals and Equations

Scalar equation of a plane: $A x + B y + C z + D = 0$ , with normal $n = (A, B, C)$ .
Through $P_{0}$ with normal $n$ : $A (x - x_{0}) + B (y - y_{0}) + C (z - z_{0}) = 0,$ so $D = - (A x_{0} + B y_{0} + C z_{0})$ .
Vector equation: $r = r_{0} + s a + t b$ , with $a$ and $b$ non-parallel vectors in the plane.
Vector to scalar: $n = a \times b$ ; three points $A$ , $B$ , $C$ : $n = A B \times A C$ .
Parallel planes: parallel normals. Perpendicular planes: $n_{1} \cdot n_{2} = 0$ .

Intersections of Lines and Planes

Line meets plane: substitute the parametric equations into the scalar equation and solve for $t$ . If $n \cdot m = 0$ the line is parallel to the plane or lies in it.
Two non-parallel planes meet in a line with direction $n_{1} \times n_{2}$ .
Three planes meet at exactly one point if and only if $n_{1} \cdot (n_{2} \times n_{3}) \neq = 0$ .
Elimination giving $0 = 0$ : infinitely many solutions (a line or a plane). Giving $0 = k$ , $k \neq = 0$ : no solution.
Two equations in 2-space usually give a point; two equations in 3-space usually give a line.

Distances in 3-Space

Point to plane: $d = \frac{∣ A x _{1} + B y _{1} + C z _{1} + D ∣}{A ^{2} + B ^{2} + C ^{2}}$ (in 2-space, $\frac{∣ A x _{1} + B y _{1} + C ∣}{A ^{2} + B ^{2}}$ ).
Parallel planes with the same $A$ , $B$ , $C$ : $d = \frac{∣ D _{1} - D _{2} ∣}{A ^{2} + B ^{2} + C ^{2}}$ .
Point to line in 3-space: $d = \frac{∣ P _{0} Q \times m ∣}{∣ m ∣}$ .
Skew lines: $d = \frac{∣ P _{1} P _{2} \cdot ( m _{1} \times m _{2} ) ∣}{∣ m _{1} \times m _{2} ∣};$ $d = 0$ means the lines intersect.
The foot of the perpendicular from $Q$ to a plane is where the line $r = OQ + k n$ meets the plane.

Where marks get lost

Using the normal as a direction vector. For $3 x - 4 y + 12 = 0$ , $(3, - 4)$ is perpendicular to the line; a direction vector is $(4, 3)$ . (Lines in 2-Space)
Giving two different lines the same parameter $t$ when finding their intersection. Each line needs its own parameter, because the meeting point usually comes at different parameter values. (Lines in 2-Space)
Mixing up the roles of the two vectors: $r = (4, 2) + t (2, - 1)$ is a line through the point $(4, 2)$ with direction $(2, - 1)$ , not a line through $(2, - 1)$ . (Lines in 2-Space)
Deciding that two vector equations describe different lines because they look different. Check whether the directions are parallel and whether a point of one satisfies the other. (Lines in 2-Space)
Calling two lines with parallel normals "parallel and distinct" without comparing the constants. If the whole equations are multiples, the lines are coincident. (Lines in 2-Space)
Writing a "scalar equation of a line" such as $2 x + 3 y - z = 4$ in 3-space. One linear equation in $x$ , $y$ and $z$ is a plane. (Lines in 3-Space)
Dividing by zero in the symmetric equations. When a direction component is $0$ , that coordinate is constant; write it separately, such as $y = 2$ . (Lines in 3-Space)
Solving two of the three component equations and announcing an intersection without checking the third. For skew lines, two of the equations usually have a solution; only the third shows that the lines never meet. (Lines in 3-Space)
Calling two lines that never meet "parallel". In 3-space they might be skew; parallel lines must have parallel direction vectors. (Lines in 3-Space)
Using the same parameter $t$ for both lines, which forces the intersection to happen at the same parameter value on each line. (Lines in 3-Space)
Including the constant in the normal: the normal of $3 x + 5 y - 2 z + 15 = 0$ is $(3, 5, - 2)$ , not $(3, 5, - 2, 15)$ . (Planes: Normals and Equations)
Getting the sign of $D$ wrong. $D = - (A x_{0} + B y_{0} + C z_{0})$ ; substitute the point into $A x + B y + C z + D = 0$ and solve rather than guessing. (Planes: Normals and Equations)
Using two parallel direction vectors (or three collinear points) for a plane. Their cross product is $0$ and they only describe a line. (Planes: Normals and Equations)
Thinking $x = 0$ means the same thing everywhere. In 2-space it is the $y$ -axis; in 3-space it is the whole $yz$ -plane. (Planes: Normals and Equations)
Assuming a vector equation of a plane is unique. Any point of the plane and any two non-parallel vectors in it give a correct equation; compare normals and test a point instead. (Planes: Normals and Equations)
Reading $n \cdot m = 0$ as "the line is perpendicular to the plane". It means the line is perpendicular to the NORMAL, so the line is parallel to the plane (or lies in it). (Intersections of Lines and Planes)
Concluding "no solution" from $n_{1} \cdot (n_{2} \times n_{3}) = 0$ . A zero triple product rules out a single point, but the planes might still share a whole line. (Intersections of Lines and Planes)
Using one of the normals as the direction of the line where two planes meet. The direction is $n_{1} \times n_{2}$ , perpendicular to both normals. (Intersections of Lines and Planes)
Setting $z = 0$ to find a point on a line of intersection whose direction has a zero $z$ component. That line is parallel to the $x y$ -plane and may never reach $z = 0$ ; set another variable instead. (Intersections of Lines and Planes)
Getting $0 = 0$ during elimination and writing "no solution". A true statement like $0 = 0$ means infinitely many solutions; a false one like $0 = 2$ means none. (Intersections of Lines and Planes)
Leaving out the absolute value and reporting a negative distance. The sign only tells you which side of the plane the point is on. (Distances in 3-Space)
Substituting into a plane written as $3 x + 5 y - 2 z = 6$ without first moving the $6$ across. In the formula $D = - 6$ , not $6$ . (Distances in 3-Space)
Using $\frac{∣ D _{1} - D _{2} ∣}{∣ n ∣}$ for $x + y + z = 1$ and $2 x + 2 y + 2 z = 5$ without first scaling one equation so the coefficients match. (Distances in 3-Space)
Using a dot product instead of a cross product for a point and a line in 3-space. $∣ P_{0} Q \times m ∣$ measures the parallelogram's area; the dot product measures the part of $P_{0} Q$ that runs ALONG the line, not across it. (Distances in 3-Space)
Applying the skew-line formula to parallel lines, where $m_{1} \times m_{2} = 0$ and the formula divides by zero. Use the point-to-line distance instead. (Distances in 3-Space)

Practice test

25 questions from across the unit. Each one shows the lesson it comes from.

Lines in 2-Space
1
Which vector is a normal to the line $5 x - 2 y + 7 = 0$ ?
Lines in 2-Space
2
A line has vector equation $r = (3, - 2) + t (1, 4)$ . What is the $y$ -coordinate of the point on the line with $x$ -coordinate $6$ ?
Lines in 2-Space
3
Which point lies on the line $x = 5 - 2 t$ , $y = 1 + 3 t$ ?
Lines in 2-Space
4
What is the slope of the line $r = (1, 2) + t (4, - 6)$ ? Give a decimal.
Lines in 2-Space
5
All four equations describe the SAME line. Match each one to the name of its form.
$3 x - 4 y + 12 = 0$
$r = (0, 3) + t (4, 3)$
$x = 4 t, y = 3 + 3 t$
$y = \frac{3}{4} x + 3$
Lines in 3-Space
6
Why can't a line in 3-space be described by one scalar equation?
Lines in 3-Space
7
The point $(10, k, 1)$ lies on the line $x = 1 + 3 t$ , $y = 2 - t$ , $z = - 5 + 2 t$ . Find $k$ .
Lines in 3-Space
8
Find the angle between the lines with direction vectors $(2, 1, - 1)$ and $(1, 2, - 3)$ , in degrees to one decimal place.
°
Lines in 3-Space
9
What are two lines in 3-space called if they are not parallel and do not intersect? (one word)
Lines in 3-Space
10
In 3-space, what is the set of points satisfying both $x = 4$ and $z = - 1$ ?
Planes: Normals and Equations
11
Which vector is a normal to the plane $3 x + 5 y - 2 z = 6$ ?
Planes: Normals and Equations
12
The point $(k, 4, 2)$ lies on the plane $2 x - y + 3 z - 6 = 0$ . Find $k$ .
Planes: Normals and Equations
13
What is the smallest number of non-collinear points that determines a unique plane?
Planes: Normals and Equations
14
Find $k$ so that the plane $k x + 2 y - z = 4$ is perpendicular to the plane $3 x - y + 4 z = 0$ .
Planes: Normals and Equations
15
Put the steps for finding the scalar equation of a plane through three points $A$ , $B$ and $C$ in order.
1. 1Check that B and C also satisfy the equation
2. 2Substitute point A to solve for D
3. 3Write Ax + By + Cz + D = 0 using that normal
4. 4Form the vectors AB and AC
5. 5Take their cross product to get a normal (A, B, C)
Intersections of Lines and Planes
16
The line $r = (1, 0, - 2) + t (2, 1, 3)$ meets the plane $x + y + z - 11 = 0$ . Find the $z$ -coordinate of the point of intersection.
Intersections of Lines and Planes
17
Three planes have normals $a$ , $b$ and $c$ with $a \cdot (b \times c) = 12$ . What can you conclude?
Intersections of Lines and Planes
18
Three planes intersect in pairs along three parallel lines, but no point lies on all three. What is this configuration called? (two words)
Intersections of Lines and Planes
19
Every point on the line where $x + y + z = 1$ meets $x - y + z = 0$ has the same $y$ -coordinate. What is it? Give a decimal.
Intersections of Lines and Planes
20
Sort these arrangements of three planes by the size of their common solution set, from smallest to largest.
1. 1Three planes whose normals are not coplanar
2. 2Three coincident planes
3. 3Three planes through one common line, like pages of a book
4. 4Three parallel, distinct planes
Distances in 3-Space
21
Find the distance from $(1, 2, 3)$ to the plane $2 x + y - 2 z + 9 = 0$ , to 2 decimal places.
Distances in 3-Space
22
Find the distance between the parallel planes $2 x - y + 2 z + 4 = 0$ and $2 x - y + 2 z - 5 = 0$ .
Distances in 3-Space
23
Find the positive value of $D$ for which the plane $2 x - y + 2 z + D = 0$ is $3$ units from the origin.
Distances in 3-Space
24
For two lines, $m_{1} \times m_{2} = 0$ . What should you do to find the distance between them?
Distances in 3-Space
25
Match each distance to the formula used to find it.
1. A. $\frac{∣ P _{0} Q \times m ∣}{∣ m ∣}$
2. B. $\frac{∣ P _{1} P _{2} \cdot ( m _{1} \times m _{2} ) ∣}{∣ m _{1} \times m _{2} ∣}$
3. C. $\frac{∣ A x _{1} + B y _{1} + C ∣}{A ^{2} + B ^{2}}$
4. D. $\frac{∣ A x _{1} + B y _{1} + C z _{1} + D ∣}{A ^{2} + B ^{2} + C ^{2}}$
Point to plane
Point to line in 2-space
Point to line in 3-space
Between skew lines

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…