Let A be a point on the line vec{r} = (1 - 3m | vedclass

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Question

(A) The coordinates of point $A$ on the line are $(1 - 3\mu, \mu - 1, 2 + 5\mu)$.The vector $\overrightarrow{AB}$ is given by $B - A = (3 - (1 - 3\mu)

Vedclass · Accepted Answer

$\frac{1}{4}$

Vedclass · Answer

(A) The coordinates of point $A$ on the line are $(1 - 3\mu, \mu - 1, 2 + 5\mu)$.The vector $\overrightarrow{AB}$ is given by $B - A = (3 - (1 - 3\mu))\hat{i} + (2 - (\mu - 1))\hat{j} + (6 - (2 + 5\mu))\hat{k}$.$\overrightarrow{AB} = (2 + 3\mu)\hat{i} + (3 - \mu)\hat{j} + (4 - 5\mu)\hat{k}$.The vector $\overrightarrow{AB}$ is parallel to the plane $x - 4y + 3z = 1$ if and only if the dot product of $\overrightarrow{AB}$ and the normal vector of the plane $\vec{n} = (1, -4, 3)$ is zero.$(2 + 3\mu)(1) + (3 - \mu)(-4) + (4 - 5\mu)(3) = 0$.$2 + 3\mu - 12 + 4\mu + 12 - 15\mu = 0$.$(3 + 4 - 15)\mu + (2 - 12 + 12) = 0$.$-8\mu + 2 = 0$.$8\mu = 2 \Rightarrow \mu = \frac{2}{8} = \frac{1}{4}$.

Let $A$ be a point on the line $\vec{r} = (1 - 3\mu)\hat{i} + (\mu - 1)\hat{j} + (2 + 5\mu)\hat{k}$ and $B(3, 2, 6)$ be a point in space. Then the value of $\mu$ for which the vector $\overrightarrow{AB}$ is parallel to the plane $x - 4y + 3z = 1$ is

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