Vectors Practice

1 Secondary 4 Mathematics: Vectors 1. Vectors • A vector is a quantity that has both magnitude and direction. • A vector can be represented by a directed line segment as shown below. • Point A is known as the starting point and point B is the ending point. • The vector above can be denoted by 𝐴𝐵#####⃗ or %𝑥 𝑦( or a. A. Magnitude of a Column Vector • For 𝐴𝐵#####⃗ = 𝒂 = %𝑥 𝑦(, magnitude of 𝒂 = |𝒂| = .𝑥! + 𝑦! B. Position Vector and Coordinates of a Point • If a point P has coordinates (a, b), the position vector of P is 𝑂𝑃#####⃗ = %𝑎 𝑏(. 2. Equal and Negative Vectors • Equal vectors are vectors with the same magnitude and direction. 𝑃𝑄#####⃗ = 𝑋𝑌#####⃗ • Negative vectors are vectors with the same magnitude but are in opposite directions. 𝑃𝑄#####⃗ = −𝑅𝑆#####⃗

2 3. Sum and Differences of Two Vectors A. Triangular Law of Addition B. Subtraction of Vectors 𝐵𝐶#####⃗ = 𝐴𝐶#####⃗ − 𝐴𝐵#####⃗ 4. Multiplying a Vector by a Scalar For parallel vectors or points that are collinear, 𝐴𝐵#####⃗ = 𝑘𝐵𝐶#####⃗. 5. Ratio of Areas of Triangles Method 1: If both triangles are similar "! "" = % #! #" ( ! Method 2: If both share a common height,

3 Vectors (Worksheet 1) 1. Find the magnitude of the following vectors: (a) 𝐴𝐵#####⃗ = %3 4( Answer: (a) ________________ units (b) 𝐵𝐶#####⃗ = %−12 −5 ( Answer: (b) ________________ units (c) 𝐶𝐷#####⃗ = % 3 −5( Answer: (c) ________________ units

4 2. Given that A is the point (−4, 6), B is the point (−1, 𝑦 − 2) and 𝐴𝐶#####⃗ = % 4 −10(, find (a) the position vectors of points A and B, Answer: (a) 𝑂𝐴#####⃗ = ________________, 𝑂𝐵#####⃗ = ________________ (b) 𝑂𝐶#####⃗ and thus the coordinates of C, Answer: (b) 𝑂𝐶#####⃗ = ________________, 𝐶 = ( _______, _______ ) (c) 𝐴𝑂#####⃗. Answer: (c) 𝑂𝐴#####⃗ = ________________ (d) Hence, using your answer in (c), find the value of y if A, B and C are collinear. Answer: (d) 𝑦 = ________________

5 3. Given that P is the point (0, 3), 𝑃𝑄#####⃗ = %−3 4 ( and R is a point on the x-axis, such that P, Q and R lies on the same straight line, find (a) D𝑃𝑄#####⃗D, Answer: (a) D𝑃𝑄#####⃗D = ________________ units (b) the coordinates of Q, Answer: (b) 𝑄 = ( _______, _______ ) R is a point on the x-axis such that point R is (x , 0). Find (c) 𝑃𝑅#####⃗ in terms of x, Answer: (c) 𝑃𝑅#####⃗ = ________________

6 Since P, Q and R lies on the same straight line, it is given that 𝑃𝑄#####⃗ = 𝑘𝑃𝑅#####⃗. Find (d) the value of k, Answer: (d) 𝑘 = ________________ (e) the position vector of R. Answer: (e) 𝑂𝑅#####⃗ = ________________

7 4. Given that A is the point (5, 2) and 𝑂𝐵#####⃗ = % 6 −3(. (a) Calculate D𝐴𝐵#####⃗D. Answer: (a) D𝐴𝐵#####⃗D = ________________ units (b) Find the equation of the line AB. Answer: (b) y = ________________

8 (c) Given that C is the midpoint of AB, find the coordinates of C. Answer: (c) 𝐶 = ( _______, _______ ) 5. In the figure below, 𝑂𝑃#####⃗ = 𝒑 and 𝑂𝑄######⃗ = 𝒒. If 𝑂𝑋#####⃗ = $ % 𝑂𝑃#####⃗, find 𝑄𝑋#####⃗ in terms of p and q. Answer: 𝑄𝑋#####⃗ = ________________

9 6. In the diagram, 𝑂𝑇 = 3𝑂𝑃, 𝑅𝑇 = 6𝑅𝑆 and 𝑄 is the midpoint of 𝑃𝑅. It is given that 𝑂𝑃#####⃗ = 𝒑 and 𝑃𝑄#####⃗ = 𝒒. (a) Express, as simply as possible, in terms of p and q, (i) 𝑂𝑅#####⃗, (ii) 𝑇𝑅#####⃗, (iii) 𝑄𝑆#####⃗. Answer: (a)(i) 𝑂𝑅#####⃗ = ________________ (a)(ii) 𝑇𝑅#####⃗ = ________________ (a)(iii) 𝑄𝑆#####⃗ = ________________

10 (b) Determine if the lines OR and QS are parallel, showing your reasons clearly. 7. In the figure below, OABC is a parallelogram. The vectors 𝑂𝐴#####⃗ = 8𝒙 and 𝑂𝐶#####⃗ = 4𝒚. The lines AB = 2BD and CE = EB. (a) Express the following vectors in terms of x and y. (i) 𝐴𝐶#####⃗, Answer: (a)(i) 𝐴𝐶#####⃗ = ________________