Name: Geometrical Interpretation of Scalar Triple Product
Uploaded: 2019-07-19
Duration: 14 min
Description: Do you know what is the geometrical meaning of the scalar triple product? And why the scalar triple product so important? Check out the video to explore.

Question

If

\vec{a} = \hat{i} + \hat{j} + \hat{k}

and

\vec{b} = \hat{j} - \hat{k}

, then find a vector

\vec{c}

such that

\vec{a} \times \vec{c} = \vec{b}

and

\vec{a} \cdot \vec{c} = 3

Accepted Answer

Given that $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$

Let $\vec{c} = x \hat{i} + y \hat{j} + z \hat{k}$

Given that

$\vec{a} \times \vec{c} = \vec{b}$

$\Rightarrow |\begin{array}{l} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 1 \\ x & y & z \end{array}| = \hat{j} - \hat{k}$

$\Rightarrow \hat{i} (z - y) - \hat{j} (z - x) + \hat{k} (y - x) = \hat{j} - \hat{k}$

Comparing corresponding vector components

$\Rightarrow z - y = 0, z - x = - 1$ and $y - x = - 1$

$\Rightarrow y = z, x - z = 1$ and $x - y = 1$

$\Rightarrow z = y . . . (i)$ and $x = 1 + y . . . . (ii)$

Also given that,

$\vec{a} \cdot \vec{c} = 3$

$\Rightarrow (\hat{i} + \hat{j} + \hat{k}) \cdot (x \hat{i} + y \hat{j} + z \hat{k}) = 3$

$\Rightarrow x + y + z = 3$

$\Rightarrow 1 + y + y + y = 3$

$\Rightarrow 3 y = 2$

$\Rightarrow y = \frac{2}{3}$

$∴ z = y = \frac{2}{3}$ and $x = 1 + y = \frac{5}{3}$

$∴ \vec{c} = \frac{5}{3} \hat{i} + \frac{2}{3} \hat{j} + \frac{2}{3} \hat{k}$

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$ , then find a vector $\vec{c}$ such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

If a→=i^+j^+k^ and b→=j^-k^, then find a vector c→ such that a→×c→=b→ and a→·c→=3

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$ , then find a vector $\vec{c}$ such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$