Vector Algebra quiz

The magnitude of 3i - 4j is:

The value of i.i is:

The value of i x j is:

If a.b = 0 for non-zero vectors a and b, then a and b are:

The value of i x i is:

The vectors 2i + 3j and 4i + 6j are:

A unit vector in the direction of i + j is:

The area of a triangle with adjacent sides a and b is:

If abs(a x b) = a.b, the angle between a and b is:

The value of i.(j x k) is:

Find the direction cosines of the vector i + 2j + 3k. (2 marks)

Find the sum of the vectors i - 2j + k, -2i + 4j + 5k and i - 6j - 7k. (2 marks)

Find a vector of magnitude 5 in the direction of 3i + 4k. (2 marks)

Find a.b if a = 2i - j + 3k and b = i + 4j - k. (2 marks)

Show that (a - b) x (a + b) = 2(a x b). (2 marks)

Find the angle between a and b if abs a = 2, abs b = 1 and a.b = 1. (2 marks)

Find a x b if a = i + j and b = j + k. (2 marks)

Find x if the vectors 2i + xj + k and i - 2j + 3k are perpendicular. (2 marks)

Write two different vectors having the same magnitude. (2 marks)

Show that the vectors 2i - 3j + 4k and -4i + 6j - 8k are collinear. (2 marks)

Find lambda if 2i + 3j + 4k is perpendicular to i + lambda j - 2k. (3 marks)

Find the area of the parallelogram whose diagonals are 3i + j - 2k and i - 3j + 4k. (3 marks)

If abs a = 2, abs b = 3 and a.b = 4, find abs(a - b). (3 marks)

Show that the points with position vectors -2i + 3j + 5k, i + 2j + 3k and 7i - k are collinear. (3 marks)

Let a = i + 4j + 2k, b = 3i - 2j + 7k and c = 2i - j + 4k. Find a vector d perpendicular to both a and b such that c.d = 15. (3 marks)

Read the passage and answer the questions. A force F = 3i + 2j - k newtons moves a block from point (1, 2, 3) to point (4, 4, 5), measured in metres. (i) Find the displacement vector. (ii) Find the work done, W = F.d. (iii) Find the angle between F and d. (5 marks)

Read the passage and answer the questions. An aeroplane flies north at 200 km/h while a wind blows east at 50 km/h. (i) Write both velocities as vectors using i (east) and j (north). (ii) Find the resultant velocity vector. (iii) Find the actual speed of the aeroplane. (5 marks)

Read the passage and answer the questions. Three towers stand at P(2, -1, 1), Q(1, -3, -5) and R(3, -4, -4). (i) Find the vectors PQ, QR and RP. (ii) Find QR.RP. (iii) Show that the towers form a right angled triangle and name the right angle. (5 marks)

Read the passage and answer the questions. A triangular garden has corners with position vectors i, j and k (units in tens of metres). (i) Find two side vectors. (ii) Find their cross product. (iii) Find the area of the garden. (5 marks)

Read the passage and answer the questions. A student says that if a.b = a.c for a non-zero vector a, then b must equal c. (i) Is the student correct? (ii) Give a counter example. (iii) What can be concluded about b - c? (5 marks)

Derive the section formula for the position vector of a point dividing a line segment internally in the ratio m:n, with a diagram. Write the formula for external division. (6 marks)

Using vectors, prove that the angle in a semicircle is a right angle. (6 marks)

Prove that abs(a + b)^2 = abs a^2 + abs b^2 if and only if a and b are perpendicular, and state the triangle inequality abs(a + b) <= abs a + abs b with a diagram. (6 marks)

Find the area of the triangle with vertices P(1, 2, 3), Q(2, -1, 4) and R(4, 5, -1) using vectors, and draw a diagram. (6 marks)

If abs a = sqrt(3), abs b = 2 and a.b = sqrt(6), find the angle between a and b and the magnitude of a x b. Draw a diagram showing both vectors. (6 marks)