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

Engineering

Mathematics

Direction Cosines and Direction Ratio

Equation of Straight Line in 3D

Distance from a Plane

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(x₁, y₁, z₁)

Direvtion ratio of line x₁− 1, y₁+ 3, z₁− 4

Since AP is parallel to

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

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

where k is any constant

⇒ x₁= 2k + 1, y₁=−3k − 3, z₁= −6k + 4

But it is given that AP = 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

P = (2k + 1, − 3k − 3,−6k + 4)

= (2 × 3 + 1,−3 × 3 − 3,−6 × 3 + 4)

= (7,−12,−14)