If A(2,1, 3), B(2, –3, 4), C(–1, 2, 7) are the three points ; find a point D such that .

Engineering

Mathematics

Vector Addition and Subtraction

Direction Cosines and Direction Ratio

theorem in space

Question

If A(2,1, 3), B(2, –3, 4), C(–1, 2, 7) are the three points ; find a point D such that $| \vec{AB} | = \frac{1}{2} | \vec{CD} | and AB ‖ CD$ .

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$\vec{AB} = \frac{1}{2} \vec{CD} or \vec{AB} = - \frac{1}{2} \vec{CD}$

$\vec{AB} = - 4 \hat{j} + \hat{k}$

$2 (- 4 \hat{j} + \hat{k}) = \vec{D} - \vec{C} 2 (- 4 \hat{j} + \hat{k}) = \vec{C} - \vec{D}$

$- 8 \hat{j} + 2 \hat{k} - \hat{i} + 2 \hat{j} + 7 \hat{k} = \hat{0} - 8 \hat{j} + 2 \hat{k} + \hat{i} - 2 \hat{j} - 7 \hat{k} = \hat{0}$

$\vec{D} = - \hat{i} - 6 \hat{j} + 9 \hat{k} - \hat{i} - 10 \hat{j} - 5 \hat{k} = - \hat{0}$

or $\vec{D} = - \hat{i} + 10 \hat{j} + 5 \hat{k}$