Question

Let $u$ and $v$ are unit vectors and $w$ is a vector such that $u\times v+u=w$ and $w\times u=v$ then find the value of $[uvw]$.

• A. $0$
• B. $1$
• C. $\left|w\right|$
• D. $2\left|w\right|$

Answer 1

The correct option is B

$1$

Explanation for the correct option:

Given that, $u\times v+u=w$

Multiplying both the sides with vector $u$

$$\begin{gathered} (u\times v+u)\times u=w\times u \\ (u\times v)\times u+u\times u=v \\ (u·u)v–(v·u)u+u\times u=v \\ \left(1\right)v–(v·u)u+0=v \\ v–(v·u)u=v \\ \Rightarrow u·v=0 \end{gathered}$$

Now,

$$\begin{gathered} u·(v\times w) \\ =u·(v\times (u\times v+u\left)\right) \\ =u·(v\times (u\times v)+v\times u) \\ =u·\left(\right(v·v)u–(v·u)v+v\times u) \\ =u\left(\right|v{|}^{2}u-0+v\times u) \\ =|v{|}^2(u·u)–u·(v\times u) \\ =|v{|}^2|u{|}^2–0 \\ =1 \end{gathered}$$

Therefore, $[uvw]=1$

Hence, the correct option is (B)