Line/plane intersections and angles | Year 12

01 / LEARN

The idea, step by step

Substitute before solving

Substitute the line $a + t d$ into $n \cdot r = c$ . If $n \cdot d \neq 0$ , solve one equation for t to find the intersection. If this coefficient is zero, test the initial point to distinguish a contained line from a disjoint parallel line.

Check a final intersection in both original objects. Two nonparallel planes intersect in a line; parallel normals require checking whether the plane constants make them coincident or distinct.

Match the angle to the geometry

For the smaller angle between planes, use $cos θ = ∣ n_{1} \cdot n_{2} ∣ / (∣ n_{1} ∣ ∣ n_{2} ∣)$ . For the smaller angle of a line with a plane, use $sin θ = ∣ d \cdot n ∣ / (∣ d ∣ ∣ n ∣)$ .

The line-plane angle is complementary to the angle with the normal. Absolute values select the smaller angle; they should not automatically be inserted when a question asks for the angle between directed vectors.

A common trap

The angle with a plane equals the angle with its normal.

Instead: These angles are complementary; use sine for the line-plane angle.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Intersect $(1, 0, 2) + t (1, 2, - 1)$ with x+y+z=6.

Substitution gives $3 + 2 t = 6$ .
Solve for t and substitute into all components.

t = 3 / 2

, point

(5 / 2, 3, 1 / 2)

Example 2

Worked example 2

Find the smaller angle of direction (1,0,1) with z=0.

A normal is $(0, 0, 1)$ .
The sine of the angle is $1 / \sqrt{2}$ .

45^{\circ}

Check your understanding

Describe $(1, 0, 0) + t (0, 1, 0)$ relative to z=0.

Show answer

The line lies in the plane.

Both its initial point and every direction displacement have z=0.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Intersect $t (1, 1, 1)$ with x+y+z=6.
2 marks
Does (0,0,1)+t(1,0,0) meet z=0?
2 marks
Find the smaller angle between x=0 and x+y=0.
2 marks
Find the smaller angle of direction (0,0,1) with z=2.
2 marks
Find the intersection line of x+y=1 and z=2.
2 marks
Find k so $(0, 0, k) + t (1, - 1, 0)$ lies in x+y+z=3.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Intersect $t (1, 1, 1)$ with x+y+z=6.
2 marks
The plane equation gives 3t=6.
Substitute t=2.
$(2, 2, 2)$
Does (0,0,1)+t(1,0,0) meet z=0?
2 marks
Its z coordinate remains 1.
It is a distinct parallel line.
No.
Find the smaller angle between x=0 and x+y=0.
2 marks
Use normals (1,0,0) and (1,1,0).
Their smaller-angle cosine is $1 / \sqrt{2}$ .
$45^{\circ}$
Find the smaller angle of direction (0,0,1) with z=2.
2 marks
The line direction is parallel to the plane normal.
It is perpendicular to the plane.
$90^{\circ}$
Find the intersection line of x+y=1 and z=2.
2 marks
Set y=t.
Then x=1-t and z=2.
$r = (1, 0, 2) + t (- 1, 1, 0)$ .
Find k so $(0, 0, k) + t (1, - 1, 0)$ lies in x+y+z=3.
2 marks
The direction contributes zero to x+y+z.
The initial point must satisfy the plane.
$k = 3$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics vector algebra, vector/Cartesian equations and linear systems. Original explanations and tasks.

Official curriculum source ↗

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