Vector Algebra

Vector Algebra deals with quantities that have both magnitude and direction, such as displacement, velocity and force, and provides the language used in physics, engineering and three dimensional geometry. In this chapter you will learn to represent vectors as directed line segments, the types of vectors (zero, unit, collinear, equal and negative vectors), and how to add vectors by the triangle and parallelogram laws. You will multiply a vector by a scalar, write vectors in component form xi + yj + zk, find magnitudes and unit vectors, and use the section formula for internal and external division. You will then study two kinds of products. The scalar or dot product, a.b = (abs a)(abs b) cos theta, gives the angle between two vectors, the condition for perpendicularity and the projection of one vector on another. The vector or cross product, whose magnitude is (abs a)(abs b) sin theta, gives a vector perpendicular to both, and is used to find the areas of parallelograms and triangles and to test whether vectors are parallel.

Detailed Worksheet: Vector Algebra Section A - Definitions (10 marks) 1. Define a unit vector and collinear vectors with one example of each. (2 marks) 2. State the triangle law of vector addition with a diagram description. (2 marks) 3. Define the scalar product of two vectors. When is it zero for non-zero vectors? (2 marks) 4. Define the vector product of two vectors. Write i x j, j x k and k x i. (2 marks) 5. Write the formula for the projection of a vector a on a vector b. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the magnitude of a = 2i + 3j - 6k and a unit vector in its direction. (3 marks) 7. The position vectors of points P and Q are i + 2j - k and -i + j + k. Find the position vector of the point dividing PQ in the ratio 2:1 (i) internally and (ii) externally. (3 marks) 8. Find the angle between the vectors i + j - k and i - j + k. (3 marks) 9. Find the projection of the vector 2i + 3j + 2k on the vector i + 2j + k. (3 marks) 10. Find the area of the parallelogram whose adjacent sides are given by 3i + j + 4k and i - j + k. (3 marks) Section C - Diagrams (10 marks) 11. Draw diagrams showing the triangle law and the parallelogram law of vector addition and show that they give the same resultant. (4 marks) 12. Draw a diagram showing the projection of vector a on vector b and mark the angle theta between them. (3 marks) 13. Draw a diagram showing vectors a and b, their cross product a x b perpendicular to both, and the parallelogram whose area equals abs(a x b). (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Find the area of the triangle with vertices P(1, 1, 2), Q(2, 3, 5) and R(1, 5, 5) using vectors. (5 marks) 15. If a = i + j + k and b = i + 2j + 3k, find a unit vector perpendicular to both a + b and a - b. (5 marks) 16. If a, b and c are vectors such that a + b + c = 0 and abs a = 3, abs b = 4, abs c = 5, find a.b + b.c + c.a. Also show that a x b = b x c = c x a. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write vectors in i, j, k form, show determinant expansion for cross products, and give exact answers.