Find the co-ordinates of a point, which is at a distance of 21 units from the point A = \left(\right. 1 , - 3 , 4 \left.\right) in the direction of vectors 2 \hat{i} - 3 \hat{j}

Engineering

Mathematics

Line of Greatest Slope

Question

Find the co-ordinates of a point, which is at a distance of

21

units from the point

A = (1, - 3, 4)

in the direction of vectors

2 \hat{i} - 3 \hat{j} - 6 \hat{k}

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Let the required point be

P (x_{1}, y_{1}, z_{1})

Direvtion ratio of line

x_{1} - 1, y_{1} + 3, z_{1} - 4

Since AP is parallel to

2 \hat{i} - 3 \hat{j} - 6 \hat{k}

So,

\frac{x_{1} - 1}{2}, \frac{y_{1} + 3}{- 3}, \frac{z_{1} - 4}{- 6} = k

where

k

is any constant

\Rightarrow x_{1} = 2 k + 1, y_{1} =, - 3 k - 3, z_{1} =, - 6 k + 4

But it is given that

A P = 21

\sqrt{{(x_{1} - 1)}^{2} + {(y_{1} + 3)}^{2} + {(z_{1} - 4)}^{2}}, = 21

\sqrt{{(2 k)}^{2} + {(- 3 k)}^{2} + {(- 6 k)}^{2}}, = 21

\sqrt{49 k^{2}}, = 21

k = 3

Therefore required point

\begin{array}{l} P = (2 k + 1, - 3 k - 3, - 6 k + 4), \\ = (2 \times 3 + 1, - 3 \times 3 - 3, - 6 \times 3 + 4), \\ = (7, - 12, - 14), \end{array}

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