:VECTORS REVIEW EXERCISE
BASIC MATHS
Past NECTA Questions on Vectors
If a = 4i + 5j and b = 6i + 9j, determine the magnitude and direction of vector v = a + b.
If a = 3, b = 7, and c = 11, evaluate |2a - 3b + c|.
Given a = 2i + 3j, b = 4i - j, and c = i + 6j, determine the unit vector in the direction of d = 2a + 3b - c.
Given vectors a = 5i - j, b = 3i + 4j, and c = 2i - 3j, calculate the resultant of a + b + c.
If u = 4i + 6j and v = 2i - 3j, find:
(a) w = 3u - 2v
(b) |w| (correct to 2 decimal places)
(c) The angle w makes with the positive x-axis (nearest degree).
If u = 3i - j, v = -2i, and w = i + 5j, find the value of |u + v + w|.
If a = i + 2j, b = 3i - j, and c = 5i + 14j, find scalars p and q such that pa + qb = c.
Vectors A = 2i + 5j and B = 4i - j are given. Find the position vector of point M, the midpoint of AB.
Given vectors a = i + 3j, b = 5i - 2j, and c = a + 4b, find the unit vector in the direction of c.
The position vectors of points A, B, and C are 4i - 3j, i + 3j, and -5i + j respectively. Find vectors AB, BC, and AC, and verify that AB + BC = AC.
Given vectors x = 3i + 2j, y = 5i - 3j, and z = 4i - 2j:
(i) Find the resultant vector r = x + y + z.
(ii) Sketch the three vectors on the same axes and label their magnitudes.
Given a = 3i, b = 7j, and c = 12j, determine:
(a) d = a + 4b - 2c
(b) Magnitude of d (leave answer in surd form).
(c) Direction cosines of d and verify that their squares sum to 1.
Find the direction cosines of c = 9i + 12j and show that the sum of their squares is 1.
If a = 2i + 3j, b = 19i - 15j, and c = 5i - 7j, find scalars x and y such that xa + yc = b.
Given vectors a = 3i + 2j, b = 8i - 3j, and c = 2i + 4j, find:
(i) d = 3a - b + 2c
(ii) A unit vector in the direction of d.
Given vectors a = 6i + 12j and b = 17i + 18j, find:
(i) c = 2a + b and its magnitude (3 sig. figs).
(ii) Sketch vector c on the x-y plane
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