First slide

Straight lines in 3D

Question

The coordinates of the point P on the line $\vec{r} = (\hat{i} + \hat{j} + \hat{k}) + λ (- \hat{i} + \hat{j} - \hat{k})$ which .n is nearest to the origin is

Moderate

A

$(\frac{2}{3}, \frac{4}{3}, \frac{2}{3})$
B

$(- \frac{2}{3}, - \frac{4}{3}, \frac{2}{3})$
C

$(\frac{2}{3}, \frac{4}{3}, - \frac{2}{3})$
D

None of these

Solution

Let the point P be (x, y, z), then the vector $x \hat{i} + y \hat{j} + z \hat{k}$ will lie on the line. Thus,
$\begin{aligned} (x - 1) \hat{i} + (y - 1) \hat{j} + (z - 1) \hat{k} = - λ \hat{i} + λ \hat{j} - λ \hat{k} \\ \Rightarrow x = 1 - λ, y = 1 + λ and z = 1 - λ \end{aligned}$
Now point P is nearest to the origin
$\begin{array}{ll} \Rightarrow D = (1 - λ)^{2} + (1 + λ)^{2} + (1 - λ)^{2} \\ or \frac{dD}{dλ} = - 4 (1 - λ) + 2 (1 + λ) = 0 \Rightarrow λ = \frac{1}{3} \end{array}$
Hence, the point $(\frac{2}{3}, \frac{4}{3}, \frac{2}{3})$

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