Name: Scalar Triple Product
Uploaded: 2019-07-19
Duration: 6 min 56 s
Description: Have you ever heard about scalar triple products? They involve both dot and cross products! Interesting, aren't they? Watch this amazing video to learn more...

Question

If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitude, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$ .

Accepted Answer

Since $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitude.

$| \vec{a} | = | \vec{b} | = | \vec{c} | = a (say)$

Also, $\vec{a} . \vec{b} = 0 = \vec{b} . \vec{c} = \vec{c} . \vec{a}$

Now $| \vec{a} + \vec{b} + \vec{c} |^{2} = (\vec{a} + \vec{b} + \vec{c})^{2} = (\vec{a} + \vec{b} + \vec{c}) . (\vec{a} + \vec{b} + \vec{c})$

$= | \vec{a} |^{2} + | \vec{b} |^{2} + | \vec{c} |^{2} + 2 (\vec{a} . \vec{b}) + 2 (\vec{b} . \vec{c}) + 2 (\vec{c} . \vec{a})$

$= a^{2} + a^{2} + a^{2} = 3 a^{2}$

$\Rightarrow | \vec{a} + \vec{b} + \vec{c} | = \sqrt{3} a$

Let $α, β and γ$ be the angles made by $\vec{a}, \vec{b} and \vec{c} with \vec{a} + \vec{b} + \vec{c}$ .

Now $\vec{a} . (\vec{a} + \vec{b} + \vec{c}) = | \vec{a} | | \vec{a} + \vec{b} + \vec{c} | \cos α$

$\Rightarrow \vec{a} . \vec{a} + \vec{a} . \vec{b} + \vec{a} . \vec{c} = a \times \sqrt{3} a \cos α$

$\Rightarrow a^{2} = \sqrt{3} a^{2} \cos α \Rightarrow \cos α = \frac{1}{\sqrt{3}}$

Now, $\vec{b} . (\vec{a} + \vec{b} + \vec{c}) = | \vec{b} | | \vec{a} + \vec{b} + \vec{c} | \cos β$

$\Rightarrow \vec{b} . \vec{a} + \vec{b} . \vec{b} + \vec{b} . \vec{c} = | \vec{b} | | \vec{a} + \vec{b} + \vec{c} | \cos β$

$\Rightarrow a^{2} = a \cdot \sqrt{3} a \cos β \Rightarrow \cos β = \frac{1}{\sqrt{3}}$

Similarly $\cos γ = \frac{1}{\sqrt{3}}$

Thus, $\cos α = \cos β = \cos γ = \frac{1}{\sqrt{3}}$

$\Rightarrow α = β = γ$

Therefore, the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b} a n d \vec{c}$

If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitude, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$ .

Important Questions on Vectors

Learn and Prepare for any exam you want !

Ifa→, b→, c→ are mutually perpendicular vectors of equal magnitude, show that the vector a→+b→+c→ is equally inclined toa→, b→ and c→.

Important Questions on Vectors

Learn and Prepare for any exam you want !

If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitude, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}, \vec{b}$ and $\vec{c}$ .