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Prove the following.
If a⃗, b⃗, c⃗ are three mutually perpendicular vectors of equal magnitude, prove that (a⃗ + b⃗ +…
If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular vectors of equal magnitude, prove that $(\vec{a} + \vec{b} + \vec{c})$ is equally inclined with vectors $\vec{a}, \vec{b}$ and $\vec{c}$.
Given: $\vec{a}, \vec{b}, \vec{c} \text{ are three mutually perpendicular vectors of equal magnitude.}$
To show: $(\vec{a} + \vec{b} + \vec{c}) \text{ is equally inclined with vectors } \vec{a}, \vec{b} \text{ and } \vec{c}$
Answer
Answer
Let $|\vec a| = |\vec b| = |\vec c| = k$ and $\vec s = \vec a + \vec b + \vec c$.
- Step 1: The vectors are mutually perpendicular, so $\vec a\cdot\vec b = \vec b\cdot\vec c = \vec c\cdot\vec a = 0$, and $\vec a\cdot\vec a = \vec b\cdot\vec b = \vec c\cdot\vec c = k^2$.
- Step 2: $|\vec s|^2 = \vec s\cdot\vec s = a^2 + b^2 + c^2 + 2(\vec a\cdot\vec b + \vec b\cdot\vec c + \vec c\cdot\vec a) = 3k^2$, so $|\vec s| = \sqrt{3}\,k$.
- Step 3: $\vec s\cdot\vec a = \vec a\cdot\vec a + \vec b\cdot\vec a + \vec c\cdot\vec a = k^2$; similarly $\vec s\cdot\vec b = k^2$ and $\vec s\cdot\vec c = k^2$.
- Step 4: If $\alpha$, $\beta$, $\gamma$ are the angles of $\vec s$ with $\vec a$, $\vec b$, $\vec c$, then $\cos\alpha = \frac{\vec s\cdot\vec a}{|\vec s||\vec a|} = \frac{k^2}{\sqrt{3}k\cdot k} = \frac{1}{\sqrt{3}}$.
- Step 5: In the same way $\cos\beta = \frac{1}{\sqrt{3}}$ and $\cos\gamma = \frac{1}{\sqrt{3}}$.
- Step 6: $\cos\alpha = \cos\beta = \cos\gamma$, so $\alpha = \beta = \gamma$. Hence $\vec a + \vec b + \vec c$ is equally inclined with $\vec a$, $\vec b$ and $\vec c$.