Position vector of P and Q are +3 -7 k and 5 -2 +4 k respectively. Then the cosine of the angle between P Q and y -axis is

Step-by-step solution

The position vectors are OP = i + 3 j - 7 k and OQ = 5 i - 2 j + 4 k . The vector PQ is given by OQ - OP = (5 - 1) i + (-2 - 3) j + (4 - (-7)) k = 4 i - 5 j + 11 k . The magnitude of PQ is | PQ | = 4^2 + (-5)^2 + 11^2 = 16 + 25 + 121 = 162 . The y-axis is represented by the unit vector j = 0 i + 1 j + 0 k . The cosine of the angle between PQ and the y-axis is given by = PQ j | PQ | | j | . Calculating the dot product: PQ j = (4)(0) + (-5)(1) + (11)(0) = -5 . Therefore, = -5 162 1 = - 5 162 . Answer: - 5 162