Question

For what value of x will the angle between the lines with direction ratios (x, 2, 4) and
(1, 0, 1) be o 45 ?

Answer 1

Answer on Question #65901 – Math – Geometry

Question

For what value of $x$ will the angle between the lines with direction ratios $(x, 2, 4)$ and $(1, 0, 1)$ be $45{}^\circ$?

Solution

Denote $a = (a_1, a_2, a_3) = (x, 2, 4)$, $b = (b_1, b_2, b_3) = (1, 0, 1)$.

According to the algebraic definition of the dot product (see [1], p.4)

$$a b = a_1 b_1 + a_2 b_1 + a_3 b_3.$$

The angle between vectors $a, b$ is defined by (see [1], p.6)

$$\cos 45{}^\circ = \frac{a \cdot b}{||a|| \cdot ||b||}.$$

Using that $\cos 45{}^\circ = \sqrt{2}/2$ and the previous formulas, we get

$$a_1 b_1 + a_2 b_1 + a_3 b_3 = ||a|| \cdot ||b|| \cdot \cos 45{}^\circ,$$ $$x \cdot 1 + 2 \cdot 0 + 4 \cdot 1 = \sqrt{x^2 + 4 + 16} \cdot \sqrt{1 + 1} \cdot \frac{\sqrt{2}}{2},$$ $$x + 4 = \sqrt{x^2 + 20},$$ $$(x + 4)^2 = \left(\sqrt{x^2 + 20}\right)^2,$$ $$x^2 + 8x + 16 = x^2 + 20,$$ $$8x = 20 - 16,$$ $$8x = 4,$$ $$x = \frac{4}{8},$$ $$x = \frac{1}{2}.$$

Answer: $x = \frac{1}{2}$.

References:

[1] S. Lipschutz; M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 978-0-07-154352-1.

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