Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.5 Junior Find the answer

Given that the direction vector of line $l\_1$ is $\overrightarrow{a}=(1,2,-2)$ , and the direction vector of line $l\_2$ is $\overrightarrow{b}=(-2,3,m)$ . If $l\_1 \perp l\_2$ , find the value of the real number $m$ $(\ \ \ \ )$ .

A: $3$
B: $2$
C: $1$
D: $\dfrac {1}{2}$

Your answer

Multiple choice: answer with the letter of the option you want.

Solution

Since $l\_1 \perp l\_2$ , their direction vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are orthogonal. Therefore, their dot product is 0.

So, we have $\overrightarrow{a} \cdot \overrightarrow{b} = 0$ . This gives us the equation:

$1 \times (-2) + 2 \times 3 - 2m = 0$

Solving this equation for $m$ , we get:

$m = 2$

Therefore, the value of the real number $m$ is $2$ .

\boxed{m = 2}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.