S2 Mathematics Practice Book · Unit 8: Parallel and orthogonal projections

Unit 8 Parallel and orthogonal projections

Key unit competence: Apply the properties of images of objects under parallel or orthogonal projections in solving related problems.

Key ideas

A parallel projection onto a line $l$ in a direction $d$ maps each point $A$ to $A^{'}$ , the point where the line through $A$ parallel to $d$ meets $l$ . Shadows made by sunlight are parallel projections.
An orthogonal projection is a parallel projection whose direction is perpendicular to the line: $A^{'}$ is the foot of the perpendicular from $A$ to $l$ .
The image of a segment $[A B]$ is the segment $[A^{'} B^{'}]$ ; the image of any figure is a segment (or a point) on the line.
Properties: points on $l$ are invariant; collinear points stay collinear; the midpoint of $[A B]$ projects to the midpoint of $[A^{'} B^{'}]$ ; ratios of lengths on the same line are kept (Thales). Lengths and angles are not kept in general.
Orthogonal projection: the image of a segment is never longer than the segment. If the segment is parallel to $l$ , the image has the same length; if it is perpendicular to $l$ , the image is a single point.
On axes: orthogonal projection onto the x-axis: $(x, y) \mapsto (x, 0)$ ; onto the y-axis: $(x, y) \mapsto (0, y)$ . A segment of length $L$ making angle $θ$ with $l$ has projection of length $L cos θ$ (studied later).

Formulas to know

(x, y) \mapsto (x, 0) (onto the x-axis)

(x, y) \mapsto (0, y) (onto the y-axis)

A^{'} B^{'} \leq A B (orthogonal)

Common misconceptions

"The image of a triangle under projection on a line is a smaller triangle." No: all images lie on the line, so the image is a segment.
"Projection keeps lengths." No: generally lengths change; only ratios of segments on the same line are kept.

Worked examples

Example 1. Find the orthogonal projection of triangle $A (1, 3)$ , $B (4, 1)$ , $C (6, 5)$ onto the x-axis.

Solution

$A^{'} (1, 0)$ , $B^{'} (4, 0)$ , $C^{'} (6, 0)$ . The image is the segment from $(1, 0)$ to $(6, 0)$ , of length 5.

Example 2. Project $P (2, 5)$ onto the x-axis in the direction of the line $y = x$ .

Solution

The line through $P$ parallel to $y = x$ is $y = x + 3$ . It meets $y = 0$ at $x = - 3$ : $P^{'} (- 3, 0)$ .

Practice questions

Easier and harder questions are mixed, just like in a real exam.

Find the orthogonal projections of $A (- 2, 3)$ , $B (1, - 4)$ and $C (4, 2)$ on the x-axis and on the y-axis.
Show answer
x-axis: $(- 2, 0)$ , $(1, 0)$ , $(4, 0)$ . y-axis: $(0, 3)$ , $(0, - 4)$ , $(0, 2)$ .
Common mistake: Setting the wrong coordinate to 0.
When is the orthogonal projection of a segment on a line equal in length to the segment?
Show answer
When the segment is parallel to the line (or lies on it).
Common mistake: Answering "always".
Explain the difference between a parallel projection and an orthogonal projection.
Show answer
In a parallel projection the projecting lines are all parallel to a chosen direction. In an orthogonal projection that direction is perpendicular to the line of projection.
Common mistake: Saying they are unrelated: orthogonal projection is a special parallel projection.
Project the square $O (0, 0)$ , $A (2, 0)$ , $B (2, 2)$ , $C (0, 2)$ onto the x-axis in the direction of $y = - x$ .
Show answer
Lines $y = - x + k$ : $O \to (0, 0)$ , $A \to (2, 0)$ , $B \to (4, 0)$ , $C \to (2, 0)$ . Image: segment from $(0, 0)$ to $(4, 0)$ .
Common mistake: Projecting orthogonally instead of along the given direction.
Find the image of $P (4, 2)$ projected onto the x-axis in the direction of $y = x$ .
Show answer
Line through $P$ : $y = x - 2$ ; at $y = 0$ , $x = 2$ : $P^{'} (2, 0)$ .
Common mistake: Using a perpendicular instead of the given direction.
A 2 m stick casts a 1.2 m shadow. A tree casts a 9 m shadow at the same time. Find the tree's height.
Show answer
$\frac{h}{9} = \frac{2}{1.2}$ , $h = 15$ m.
Common mistake: Inverting the ratio.
Segment $A B$ has $A (1, 1)$ and $B (7, 9)$ . Find the length of $A B$ and of its orthogonal projections on both axes.
Show answer
$A B = 36 + 64 = 10$ ; on the x-axis 6, on the y-axis 8.
Common mistake: Expecting the projections to add to 10.
List three properties of orthogonal projection on a line.
Show answer
Points on the line are invariant; the image of a segment is shorter than or equal to it; midpoints map to midpoints (and collinearity is kept).
Common mistake: Claiming angles are preserved.

Matching and drag-and-drop

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. Projecting all points of a circle orthogonally onto a diameter's line gives:	a. Collinear points stay collinear
2. Which property is kept by any parallel projection onto a line?	b. the diameter (a segment)
3. Project $P (1, 5)$ onto the y-axis parallel to the direction $(21)$ .	c. $(2.5, 2.5)$
4. Find the orthogonal projection of $A (2, 3)$ onto the line $y = x$ .	d. $(0, 4.5)$

Show answer

1 → b (the diameter (a segment)) · 2 → a (Collinear points stay collinear) · 3 → d ( $(0, 4.5)$ ) · 4 → c ( $(2.5, 2.5)$ )

Every point projects between the ends of the diameter.
Images of collinear points lie on the same line.
Step 1. Points on the projecting line: $(1 - 2 t, 5 - t)$ .
Step 2. On the y-axis $1 - 2 t = 0$ , so $t = 0.5$ and the image is $(0, 4.5)$ .
Step 1. Move from $A$ along the perpendicular direction $(1, - 1)$ : $(2 + t, 3 - t)$ .
Step 2. On $y = x$ : $2 + t = 3 - t$ , so $t = 0.5$ and the image is $(2.5, 2.5)$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
the midpoint of the image segment · an endpoint · 3 · $(- 4, 0)$
1. The orthogonal projection of $(- 4, - 2)$ onto the x-axis is .
2. The orthogonal projection of segment $A (1, 4)$ , $B (5, 1)$ onto the y-axis has length .
3. Under parallel projection, the midpoint of a segment is mapped to .
Show answer
1: $(- 4, 0)$ · 2: 3 · 3: the midpoint of the image segment
- Keep $x = - 4$ , set $y = 0$ .
- Images $(0, 4)$ and $(0, 1)$ .
- Ratios on a line are preserved (Thales).

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. $A$ , $M$ , $B$ are on a line with $A M = 2$ , $MB = 4$ . Their parallel projections satisfy $A^{'} M^{'} : M^{'} B^{'} =$	a. 5 cm
2. Project $P (4, 1)$ onto the x-axis parallel to the direction $(11)$ .	b. $1 : 2$
3. A segment 10 cm long makes an angle of $6 0^{\circ}$ with line $l$ . Find the length of its orthogonal projection on $l$ .	c. $(3, 0)$
4. Find the orthogonal projection of $(3, 4)$ onto the line $y = 2 x$ .	d. $(2.2, 4.4)$

Show answer

1 → b ( $1 : 2$ ) · 2 → c ( $(3, 0)$ ) · 3 → a (5 cm) · 4 → d ( $(2.2, 4.4)$ )

Ratios on a line are preserved.
Step 1. Points on the projecting line: $(4 - t, 1 - t)$ .
Step 2. On the x-axis $1 - t = 0$ , so $t = 1$ and the image is $(3, 0)$ .
Step 1. Projected length $= 10 cos 6 0^{\circ}$ .
Step 2. $10 \times \frac{1}{2} = 5$ cm.
Step 1. The image is $t (1, 2)$ with $(3 - t, 4 - 2 t) \cdot (1, 2) = 0$ : $3 - t + 8 - 4 t = 0$ , so $t = \frac{11}{5}$ .
Step 2. Image $= (\frac{11}{5}, \frac{22}{5}) = (2.2, 4.4)$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
$(0, 7)$ · 2 m · $(3, 0)$ · $(7, 0)$
1. The orthogonal projection of $(3, 7)$ onto the y-axis is .
2. The orthogonal projection of $(3, 7)$ onto the x-axis is .
3. A 6 m pole casts a shadow of 8 m. At the same time a 1.5 m stick casts a shadow of .
Show answer
1: $(0, 7)$ · 2: $(3, 0)$ · 3: 2 m
- Keep $y$ , set $x = 0$ .
- Drop a perpendicular to the x-axis: keep $x$ , set $y = 0$ .
- Parallel projection keeps ratios: $\frac{1.5}{6} \times 8 = 2$ .

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