The foot of the perpendicular from P(5, 7, 3) | vedclass

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(D) Let $A = (9, 13, 15)$,$B = (12, 21, 10)$,and $P = (5, 7, 3)$. Let $Q(x, y, z)$ be the foot of the perpendicular from $P$ to the line $AB$.The dire

Vedclass · Accepted Answer

$(9, 13, 15)$

Vedclass · Answer

(D) Let $A = (9, 13, 15)$,$B = (12, 21, 10)$,and $P = (5, 7, 3)$. Let $Q(x, y, z)$ be the foot of the perpendicular from $P$ to the line $AB$.The direction ratios of line $AB$ are $(12 - 9, 21 - 13, 10 - 15) = (3, 8, -5)$.The equation of line $AB$ is $\frac{x - 9}{3} = \frac{y - 13}{8} = \frac{z - 15}{-5} = \lambda$.Any point $Q$ on the line $AB$ is given by $Q = (3\lambda + 9, 8\lambda + 13, -5\lambda + 15)$.The direction ratios of $PQ$ are $(3\lambda + 9 - 5, 8\lambda + 13 - 7, -5\lambda + 15 - 3) = (3\lambda + 4, 8\lambda + 6, -5\lambda + 12)$.Since $PQ \perp AB$,the dot product of their direction ratios is zero:$3(3\lambda + 4) + 8(8\lambda + 6) - 5(-5\lambda + 12) = 0$$9\lambda + 12 + 64\lambda + 48 + 25\lambda - 60 = 0$$98\lambda = 0 \implies \lambda = 0$.Substituting $\lambda = 0$ into the coordinates of $Q$,we get $Q = (9, 13, 15)$.Thus,the foot of the perpendicular is $(9, 13, 15)$,which corresponds to option $D$.

The foot of the perpendicular from $P(5, 7, 3)$ to the line joining $A(9, 13, 15)$ and $B(12, 21, 10)$ is:

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