Vectors 1

1.

Two vectors a and b are shown above. Show in a diagram a + b, a – b and b – a

If a has components (3, 4, -1) and b (2, 4, 1), find the components of a + b, a – b and b – a

2. Show that P(3,4,-1), Q(-9,-2,3) and R(9,8,11) are vertices of an isoceles triangle.

3. If , and represent the vectors 2 1 and -3 respectively,

-1 -4 5

-3 2 1

show that S coincides with P, and that triangle PQR is right angled.

4. A point P lies on the line AB. Find the coordinates of P if:

a) A is (1,0,2) and B is (5,4,10) and AP:PB = 3:1

b) A is (3,-2,0) and B is (3,-2,3) and AP:PB = 3:2

5. P is the point ( ), Q is (1,0,0) and R is (2,5,2). Show that P,Q, and R are collinear, and find S such that (be careful of the directions of the vectors)

6. The vertices of a triangle are P(-2,-2,-2), Q(1,-2,1) and R(1,0,-1). S divides PQ in

the ratio 2:1

a) Find the coordinates of S

b) Show that triangle QSR is right angled

7. The points A(-1,5,4), B(2,-1,-2) and C(3,p,q) are collinear.

a) Find the ratio in which B divides AC

b) Find p and q.