Four vectors (r = alpha i + beta j), where (alpha text{ and } beta) are constants, (s = 2i -j, m = 3i + 2j) and (n = i + j) are such that the magnitude of r is three times as s and is parallel to the vactor (m – n).
(a) Find the values of (alpha) and (beta).
(b) Calculate the magnitude and direction of (r – s).
Explanation
(a) (r = alpha i + beta j ; s = 2i – j)
(|r| = sqrt{alpha^{2} + beta^{2}} ; |s| = sqrt{2^{2} + (-1)^{2}} = sqrt{5})
(|r| = 3|s| = 3sqrt{5})
(implies alpha^{2} + beta^{2} = (3sqrt{5})^{2} = 45 …. (1))
(m = 3i + 2j ; n = i + j)
((m – n) = (3i + 2j) – (i + j) = 2i + j)
(r) is parallel to ((m – n)), hence, they both have the same gradient.
(frac{beta}{alpha} = frac{1}{2} implies alpha = 2beta)
Put (alpha = 2beta) in (1),
((2beta)^{2} + beta^{2} = 45 implies 5beta^{2} = 45)
(beta^{2} = 9 implies beta = 3)
(alpha = 2beta = 2(3) = 6)
(r = 6i + 3j).
(b) (r = 6i + 3j ; s = 2i – j)
((r – s) = (6i + 3j) – (2i – j) = 4i + 4j)
(|r – s| = sqrt{4^{2} + 4^{2}} = sqrt{32} = 4sqrt{2})
(Direction : tan theta = frac{4}{4} = 1)
(theta = 45°)
(r – s) makes an angle of 45° with the x- axis.