Cape Vectors Work Sheet

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  • Topic: Dot product, Vectors, Linear algebra
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CAPE – Pure Mathematics – Unit 1

Find the magnitude and direction of the vectors – 3i + 4j, -5i + 12j, -10j, i – j

The vector XY has magnitude 10 units and is inclined at 300 to the x-axis. Express XY as a column vector

The vector PQ has magnitude 5 units and is inclined at 1500 to the x-axis. Express PQ as a column vector

Find numbers m and n such that m 35 + n 21= 49

If a = 38+ λ and b = λ3 where a and b are parallel, find λ.

The position vectors of points A and B are 2-1 and 32 respectively. Find the unit vector in the direction of BA.

If a = 3i + j and b = -i + 3j, show that a and b are perpendicular vectors.

If 3λand 2λ2-1 are perpendicular, find λ.

ABCD is a parallelogram with vertices A(-2, -1), B(4, 1), C(3, 5) and D(p, q) (a) Find the value of p and the value of q.
(b) Find the angle between AB and AD.

P and Q are points (3, 5) and (m, 7) respectively and O is the origin. Given that cos∠ POQ= 334 find the value of m.
The position vectors of points A, B and C relative to the origin O are 2i + 3j, 10i + 2j and λ(-i + 5j) respectively. (a) Obtain an expression in terms of λ for (i) AC (ii) AC (b) Given that AB = 2AC show that 104λ2- 104λ-13=0.

(c) Evaluate the scalar product AB. AC and find the value of λ such that AB. AC=0.

a = 3i + 5j and b = -4i + 2j. Find the angle between a and b.

If a8 is parallel to 24 find the value of a.

Find a vector of magnitude 35 units and parallel to 2i – j.

If the point P has position vector -5i + 3j and Q is a point such that PQ = 7i – j, find the position vector of Q.

The three points A, B and C have position vectors i – j, 5i – 3j and 11i – 6j respectively. Show that A, B and C are collinear.

If the angle between the vectors a = λ2 and b = 31 is 450, find λ.

If a = xy and b = 21 write down a relationship between x and y for each of the following (a) a is of magnitude 5 units
(b)a is perpendicular to b
(c)a is...
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