Show that $u · (u × v) = 0$

Question

I'm unsure of how to show the following without using an example which I presume isn't the correct to show this.

The question is

Let $u = (u_1, u_2, u_3)$ and $v = (v_1, v_2, v_3)$ where $u$ & $v$ are vectors in $\Bbb{R}^3$.

Show that $u · (u × v) = 0$.

I understand how to do the dot and cross product of vectors, however, I'm not sure how to show that this equals $0$.

Answer 1

Hint. $x\times y$ is perpendicular (orthogonal) to both $x$ and $y$. If $x$ and $y$ are perpendicular (orthogonal), then $x\cdot y=0$. This is true of any vectors $x,y\in\mathbb R^3$.

Answer 2

First, calculate $x=u\times v$. That is, calculate $x_1,x_2,x_3$ such that $(x_1,x_2,x_3)=u\times v$.

Second, calculate $u\cdot x$, that is, $u\cdot x = u_1x_1+u_2x_2+u_3x_3$ and prove it is $0$.

Answer 3

The triple product of three vectors in $\Bbb R^3$ can be written as a determinant: $$ {\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\det {\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{bmatrix}}={\rm {det}}\left(\mathbf {a} ,\mathbf {b} ,\mathbf {c} \right).} $$ This implies that $\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c}) = 0$ if any two of the three vectors are equal. This is also obvious from the geometric interpretation as the (signed) volume of a parallelepiped.

The triple product does also not change if the operands are shifted in circular order: $$ {\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )=\mathbf {c} \cdot (\mathbf {a} \times \mathbf {b} )} $$ so if you know that the cross product of a vector $\mathbf {u}$ with itself is the zero vector then $$ \mathbf {u} \cdot (\mathbf {u} \times \mathbf {v} ) = \mathbf {v} \cdot (\mathbf {u} \times \mathbf {u} ) = \mathbf {v} \cdot \mathbf {0} = 0 \, . $$

Generally, the triple product of three vectors in $\Bbb R^3$ is zero if and only if the vectors are linearly dependent (as can be seen from the determinant formula).

Answer 4

Here, u×v will give vector which have direction perpendicular to both u and v ,so after this cross multiplication resultant vector's dot product with any of u or v will give zero.