QCE Specialist Maths: Vector equations: intersection of planes Q166

Q166 · Practice questionTechnology-freeComplex familiar6 marks

QUESTION 166 (6 marks)

Two planes have Cartesian equations

\Pi_1:x+y+z=6,\qquad\Pi_2:x-y+z=2.

Their line of intersection is

L

. The diagram is schematic.

Schematic view of two intersecting planes Pi1 and Pi2. Their shared line L is drawn with a solid visible portion and dashed extension.

Use a cross product to determine a direction vector for

L

[2 marks]

Determine a vector equation for

L

[2 marks]

Determine the acute angle between the planes, exactly and to two decimal places in degrees.

[2 marks]

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WORKED SOLUTION

Practice marking scheme

6 marks

ANSWER

(a)

\mathbf d=(1,0,-1)

is a direction vector. (b)

\mathbf r=(0,2,4)+t(1,0,-1)

t\in\mathbb R

. (c) Acute angle

\cos^{-1}(1/3)\approx70.53

degrees.

Worked solution

(a) Normal vectors are

\mathbf n_1=(1,1,1)

and

\mathbf n_2=(1,-1,1)

. Their cross product is

(2,0,-2)

, so

(1,0,-1)

is a direction vector for the intersection.

(b) Subtract the equations to obtain

2y=4

, hence

y=2

. Either plane then gives

x+z=4

. Set

x=t

z=4-t

, so

\mathbf r=(0,2,4)+t(1,0,-1),\quad t\in\mathbb R.

Both plane equations hold for every real

t

\cos\alpha=\frac{|\mathbf n_1\cdot\mathbf n_2|}{|\mathbf n_1||\mathbf n_2|}=\frac1{\sqrt3\sqrt3}=\frac13.

Thus

\alpha=\cos^{-1}(1/3)\approx70.53

degrees. The direction vector of the intersection is perpendicular to both normals.

Equivalent mathematically justified methods accepted.

Exact answers unless a decimal accuracy is specified.

Part a: Identify both normal vectors.

[1 mark]

Part a: Compute a nonzero cross product parallel to (1,0,-1).

[1 mark]

Part b: Find a point on both planes.

[1 mark]

Part b: Give a correct vector equation and real parameter domain.

[1 mark]

Part c: Use the normal-vector scalar product to obtain cosine 1/3.

[1 mark]

Part c: Give arccos(1/3) and 70.53 degrees.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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Vector equations: intersection of planes — Question 166

QUESTION 166 (6 marks)

Practice marking scheme