A BC D is parallelogram. The position vectors of A and C are respectively, 3 i ^ + 3 j ^ + 5 k ^ and i ^ − 5 j ^ − 5 k ^ . If M is the midpoint of the diagonal D B , then the magnitude of the proj

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A BC D is parallelogram. The position vectors of A and C are respectively, 3 i ^ + 3 j ^ ​ + 5 k ^ and i ^ − 5 j ^ ​ − 5 k ^ . If M is the midpoint of the diagonal D B , then the magnitude of the projection of OM on OC , where O is the origin, is

TestHub · Accepted Answer

In a parallelogram, diagonals bisect each other. So, mid point of $D B$ is also the midpoint of $A C$ . Mid-point of $M = 2 \hat{i} - \hat{j}$ Direction ratio of $OC = (1, - 5, - 5)$ Direction ratio of $OM = (2, - 1, 0)$ Angle $θ$ between $OM$ and $OC$ is given by $cos θ = \frac{( 1 \times 2 + ) - 5 ( - 1 ) ( + - 5 ) 0 (}{2 ^{2} + ( - 1 ) ^{2} ( ) ^{2} + ( - 5 ) ^{2} + ( - 5 ) ^{2}}) ()$ $= \frac{2 + 5}{5 51} = \frac{7}{5 51}$ Projection of $OM$ on $OC$ is given by $∣ OM ∣ \cdot cos θ = 5 \times \frac{7}{5 \times 51} = \frac{7}{51}$

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