1
$\begingroup$

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 $0$.

$\endgroup$
  • 4
    $\begingroup$ Step 1: write down the definition of $u\times v$ Step 2: write down inner product of $u$ and $u\times v$. This is something that you can do yourself. $\endgroup$ – Kavi Rama Murthy Oct 8 '19 at 7:28
2
$\begingroup$

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 $x,y\in\mathbb R^3$.

$\endgroup$
2
$\begingroup$

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 $0$.

$\endgroup$
2
$\begingroup$

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).

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.