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IB Mathematics SL Review, Vectors 1

This worksheet is due on Thurs., Nov. 15 (1st and 5th pd) or Fri., Nov. 16 (6th pd). Work all problemson your own paper. Problem 2 also requires you to draw on the question sheet, so you will attach the

question sheet to your other responses when you turn it in. Wherever possible, justify your answers by

showing work. Numerical answers should be given exactly or to three significant figures. The

assignment counts 25 points.

1. (no calculator)ABCD is a rectangle and O is the midpoint of [AB].

Express each of the following vectors in terms of OC

and OD

.(a) CD

(b) OA

(c) AD

2. (no calculator) The triangle ABC is defined by the following information

2

3OA

=

,3

4AB

=

, AB BC

= 0, AC

is parallel to0

1

.

(a) On the grid below, draw an accurate diagram of triangle ABC.

(b) Write down the vector OC

.

3. (no calculator) The vectors i,jare unit vectors along thex-axis andy-axis respectively. Thevectors u = i + 2j and v = 3i + 5j are given.

(a) Find u + 2v in terms ofi andj.

A vector w has the same direction as u + 2v, and has a magnitude of 26.

(b) Find w in terms ofi andj.

A B

CD

O

O

1

2

3

4

2 1

4

3

2

1

1 2 3 4 5 6

y

x

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4. The quadrilateral OABChas vertices with coordinates O(0, 0),A(5, 1),B(10, 5) and C(2, 7).

(a) Find the vectors OB

and AC

.(b) Find the angle between the diagonals of the quadrilateral OABC.

5. (no calculator) The vectors2

3

x

x

and

1

5

x +

are perpendicular for two values ofx.

(a) Write down the quadratic equation which the two values ofx must satisfy.(b) Find the two values ofx.

6. (no calculator) Consider the vectors c = 3i + 4j and d = 5i 12j.

Calculate the scalar productc d.

7. (a) Find the scalar product of the vectors60

25

and30

40

.

(b) Two markers are at the points P (60, 25) and Q (30, 40). A surveyor stands at O (0, 0)and looks at marker P. Find the angle she turns through to look at marker Q.

8. (no calculator) The points A and B have the position vectors2

2

and3

1

respectively.

(a) (i) Find the vector AB

.

(ii) Find AB

.

The point D has position vector23

d

.

(b) Find the vector AD

in terms ofd.

The angle BAD is 90.(c) (i) Show that d= 7.

(ii) Write down the position vector of the point D.The quadrilateral ABCD is a rectangle.

(d) Find the position vector of the point C.

(e) Find the area of the rectangle ABCD.

9. A triangle has its vertices at A(1, 3), B(3, 6) and C(4, 4).

(a) Show that AB AC

= 9.

(b) Show that, to three significant figures, cos BAC = 0.569.

10. (no calculator) The lineL passes through A (0, 3) and B (1, 0). The origin is at O. The point

R (x, 3 3x) is onL, and (OR) is perpendicular toL.

(a) Write down the vectors AB

and OR

.(b) Use the scalar product to find the coordinates of R.