A triangle is specified by the coordinates of its vertices A (3, 2, -3), B (5, 1, -1), and C (1, -2, 1). Calculate the coordinates of the vector a which is of the same direction as the vector $- - \to A B$ and has the same length as the vector $- - \to A C .$

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Solution

$\to O A = 3^i + 2^j - 3^k$

$\to O B = 5^i + 1^j - 1^k$

$\to O C = 1^i - 2^j + 1^k$

$\to A B = \to O B - \to O A = (5^i + 1^j - 1^k) - (3^i + 2^j - 3^k)$

$\to A B = 2^i - 1 \to j + 2^k$

$^A B = \frac{2^i - 1 \to j + 2^k}{3} \Rightarrow d i r e c t i o n o f \to A B$

$\to A C = \to O C - \to O A = (1^i - 2^j + 1^k) - (3^i + 2^j - 3^k)$

$\to (A C) = - 2^i - 4^j + 4^k$

$| \to A C | = \sqrt{(- 2)^{2} + (- 4)^{2} + 4^{2}} = \sqrt{36} = 6$

$n e w v e c t o r = | \to A C |^A B$

$= 6 \times \frac{2^i - 1 \to j + 2^k}{3}$

$= 4^i - 2^j + 4^k$

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