Q. 72

Question

Let $u, v$ , and $w$ be three mutually perpendicular vectors in $ℝ^{3}$ .

(a) Prove that $u \times (v \times w) = 0$ .

(b) Show that $| u \cdot (v \times w) | = ‖ u ‖ ‖ v ‖ ‖ w ‖$ .

Step-by-Step Solution

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Answer

Ans:

part (a).

$u \times (v \times w) = (‖ u ‖ ‖ w ‖ \cos 90 °) v - (‖ u ‖ ‖ v ‖ \cos 90 °) w = (0) v - (0) w = 0$

part (b)

. $u \cdot (v \times w) = ‖ u ‖ ‖ v \times w ‖ \cos θ = ‖ u ‖ ‖ v \times w ‖ \cos 0 ° = ‖ u ‖ ‖ v \times w ‖ = ‖ u ‖ ‖ v ‖ ‖ w ‖ \sin 90 ° = ‖ u ‖ ‖ v ‖ ‖ w ‖$

1Step 1. Given information:

There are three mutually perpendicular vectors in $ℝ^{3}$ .

$u \times (v \times w) = 0$
$| u \cdot (v \times w) | = ‖ u ‖ ‖ v ‖ ‖ w ‖$

2Step 2. Solving part (a):

The equation $u \times (v \times w) = (u \cdot w) v - (u \cdot v) w$ gives:

$u \times (v \times w) = (‖ u ‖ ‖ w ‖ \cos 90 °) v - (‖ u ‖ ‖ v ‖ \cos 90 °) w (Vectors are perpendicular) = (0) v - (0) w (Simplify) = 0$

Therefore, if three vectors are mutually perpendicular then $u \times (v \times w) = 0$ holds.

3Step 3. Solving part (b).

The vectors $u$ and $v \times w$ are parallel to each other.

The value of $u \cdot (v \times w)$ is:

$u \cdot (v \times w) = ‖ u ‖ ‖ v \times w ‖ \cos θ = ‖ u ‖ ‖ v \times w ‖ \cos 0 ° (Angle between u and v \times w) = ‖ u ‖ ‖ v \times w ‖ = ‖ u ‖ ‖ v ‖ ‖ w ‖ \sin 90 ° (Because vectors v and w are perpendicular) = ‖ u ‖ ‖ v ‖ ‖ w ‖$

Therefore, the result $u \cdot (v \times w) = ‖ u ‖ ‖ v ‖ ‖ w ‖$ is proved.

Q. 71

Q. 73

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