Find the values of a and b so that the points \left(\right. a , b , 3 \left.\right) , \left(\right. 2 , 0 , - 1 \left.\right) and \left(\right. 1 , - 1 ,

Engineering

Mathematics

Line of Greatest Slope

Question

Find the values of

a

and

b

so that the points

(a, b, 3), (2, 0, - 1)

and

(1, - 1, - 3)

are collinear.

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Let

A (a, b n, 3) B (2, 0, - 1) C (1, - 1, - 3)

Given they are collinear

O A = a \hat{i} + b \hat{j} + 3 \hat{k} O B = 2 \hat{i} - \hat{k} O C = \hat{i} - \hat{j} - 3 \hat{k}

\overline{A B} = O B - O A = (2 - a) \hat{i} + (1 - b) \hat{j} + (- 1 - 3) \hat{k} = (2 - a) \hat{i} - b \hat{j} - 4 \hat{k} - - - (1)

\overline{B C} = O C - O B = (1 - 2) \hat{i} + (- 1 + 0) \hat{j} + (- 1 + 1) \hat{k} = - \hat{i} - \hat{j} - 2 \hat{k} - - - (2)

As given point are collinear so they are parallel

from

(1)

(2) \frac{2 - a}{- 1} = \frac{- b}{- 1} = \frac{- 4}{- 2}

\frac{2 - a}{- 1} = 2 \frac{- b}{- 1} = 2 ∴ a = 4

2 - a = - 2 b = 2 b = 2

- a = - 4

a = 4

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