# 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

**(Marks available: 6)**

**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__ =

__+ λ__

*a*

*b*Where __ a__ is a point on the line,

__is a vector parallel to the line and λ is any number.__

*b*__ a__ = the first point

*A*.

__ b__ = point

*B*minus point

*A*.

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

**(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

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

**(Marks available: 3)**

**Answer outline and marking scheme for question: 2**

**Give yourself marks for mentioning any of the points below:**

At *t* = 0, the vectors equals

The magnitude of the velocity equals

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:**

and

**a)** determine whether these two lines intersect

**b)** find the angle between them.

**(Marks available: 6)**

**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,

=

= -0.855

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

**(3 marks)**

**(Marks available: 6)**