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Exam Style C4 Vectors Questions - Solutions

1) a) We use the equation

(As )

(Simplifying the direction vector)

or written as

b) To find the coordinates where OC lies to r, we use the direction component of the vector equation r. Use

(Expanding and solving for λ)

Therefore, the coordinates of C are

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c) It is helpful to draw a diagram to visualise how best to answer the question:

We can see that the vector is the same as the vector .

Thus

d) Again, drawing a diagram can help us visualise the best way to find the area.

One can see that the area of the parallelogram is the same as the square of length and height . Thus the Area = .

Thus Area = .

r . .

.

B

O

A

D

r

. O

A

D

. . B C

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2) a) By definition, the points A and B must lie on the line r.

Thus point B lies on r.

By comparing the coefficients of i, j & k we get the equations:

[1] [2]

[3] Using equation [3] we can solve for λ.

Using this in equations [1] and [2] we find that:

b) If point P is such that , then we use

We dot the vector equation for r with the direction component of r.

(Substituting back into vector equation for r)

Thus

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c) Drawing a diagram:

The Area of a triangle is . Thus:

Thus:

r

. O

B

A P

. .

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3) a) At point B the vector equations for the lines are equal. Thus:

(This leads us to the equations)

[1]

[2] [3]

Generally, one solves equations [1] & [2] simultaneously and substitute into equation [3] to find the values of λ and µ. Fortunately, this question has an easy route. Taking equation [3]:

Substituting this into the line equation for l1.

b) To find the angle, we use the equation:

It is important to note that the vectors that are dotted together must be the direction components of the lines only.

Let a equal the direction component of l1,

Let b equal the direction component of l2,

Continued overleaf...

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Thus

c) (Substituting the vectors for a and b)

(Substituting the vectors for c and b)

Since both and equal , .

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d) This question is remarkably easy to answer and the technique used to answer

this question arises often. But as always, it is a good idea to draw a diagram.

From the diagram above, it is easy to see that the position vector of D or if one prefers is simply:

Thus,

l1

.

O

C D

B A

l2

. .

.

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4) a) We are given the vectors:

&

Given ,

b) If OACB is a rectangle, then it means that:

(As they meet at a right angle) &

(As they also meet at a right angle) We can find & :

Thus trying :

(Thus they are at right angles)

(Thus they are at a right angle)

Thus the exact area would be :

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c)

d) We need to find the vectors & .

Thus the angle would be:

˚

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5) a) The line is given by the equation

Thus:

b) One can write line l2 as:

If point A lies on l2 then:

Be equating coefficients we arrive at three equations:

1) 2) 3)

Since for all three coefficients, point A exists on the line l2.

c) Using the formula

Using only the direction components of the two lines:

Continued overleaf...

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˚

d) It is often best to draw a diagram to visualise how best to answer this question.

There are many ways in which one can answer this question, but the best way to answer this question is to use trigonometry. Since the shortest line linking point C to the line l1 will meet the line l1 at a right angle, we have a right angle triangle. Thus: Distance = (Thus determining )

units

Distance = Distance = 1.00 units

l2

O

A

C

l1

. . . Distance

19.5 ˚

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6) a) Lines l1 and l2 meet at point Q such that:

Be equating coefficients we arrive at three equations:

1) 2) (Solving Equations 1 and 2) 3)

(Substituting into Equation 2)

To confirm that the two lines actually do intersect, we test these two values in Equation 3.

Since the above equation is true, we have confirmed that the two lines do in fact, intersect. To find the coordinates, we substitute either µ or λ:

Continued overleaf...

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b) If two vectors are perpendicular,

Taking only the direction components:

Thus, the two lines are perpendicular.

c) It is best to draw a diagram to visualise the problem:

The area of the triangle is: Area =

To find the full coordinates of P we need to determine the value of λ:

Therefore:

To find the full coordinates of R we need to determine the value of µ.

Continued overleaf...

l2

O

Q

R

l1

. .

. P

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Therefore:

Substituting in these values:

END

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