Exam-style Questions: Vectors, Lines and Planes

1. The line L passes through the points A (3, 0, -1) and B (5, -1, 4).

a) Find the vector equation of the line L.

b) Determine whether or not the line L intersects the line with the equations

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(Marks available: 6)

Answer

Answer outline and marking scheme for question: 1

Give yourself marks for mentioning any of the points below:

a) The equation for a line should be expressed as:

r = a + λb

Where a is a point on the line, b is a vector parallel to the line and λ is any number.

a = the first point A.

b = point B minus point A.

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Putting these values into the equation of the line above, gives:

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(2 marks)

b) Consider the point where the x values are the same for both lines, therefore:

3 + 2λ = 5 - 4μ

Consider the point where the y values are the same for both lines, therefore:

0 - 1λ = 1 + 1μ

Solving these equation simultaneously, gives:

λ= -3, μ = 2.

Putting these values into equation for line L, gives:

z = -1 + 5λ = -16

Putting these values into equation for line r, gives:

z = 11 + 3μ= 17.

As the value of z is not the same, both the line cannot be at the same point in space (i.e. they do not intersect).

(4 marks)

(Marks available: 6)

2. A body of mass 0.5 kg moves so that its velocity at time t seconds is

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Find the magnitude of the momentum when t = 0 and t = 2.

(Marks available: 3)

Answer

Answer outline and marking scheme for question: 2

Give yourself marks for mentioning any of the points below:

At t = 0, the vectors equals

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The magnitude of the velocity equals

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Therefore the momentum at t = 0 equals

0.5 x 12.17 = 6.08 kgms-1.

Performing the same calculation at t = 2, gives the momentum equal to

0.5 x 4.47 = 2.23 kgms-1.

(Marks available: 3 marks)

3. Two lines A and B, have the following formulas:

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and

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a) determine whether these two lines intersect

b) find the angle between them.

(Marks available: 6)

Answer

Answer outline and marking scheme for question: 3

Give yourself marks for mentioning any of the points below:

a) Matching the x-values gives: 4 - 4λ = 6 +2μ

Matching the y-values gives: 0 + 8λ = -10 - 6μ

Matching the z-values gives: -2 -2λ = -10 -2μ

Solving the first two simultaneous equations gives: λ = 1, μ = -3.

These values work in the third equation therefore the lines meet.

Substituting λ = 1, μ = -3 into the equation of lines gives the point of intersection as being:

x = 0, y = 4, z = -2.

Therefore the lines meet at (0, 4, -2)

(3 marks)

b) The angle between the lines is the angle between the direction vectors, so using the scalar product we get,

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= Copyright S-cool

= -0.855

This gives θ = 148.8o (or 180o - 148.8o = 31.2o).

(3 marks)

(Marks available: 6)