Question

The direction cosines of the vector $\hat{i}-5\hat{j}+8\hat{k}$ are

• A. $(\frac{1}{3},\frac{-5}{3},\frac{8}{3})$
• B. $(\frac{1}{\sqrt{10}},\frac{-5}{\sqrt{10}},\frac{8}{\sqrt{10}})$
• C. $(\frac{1}{3\sqrt{10}},\frac{-5}{3\sqrt{10}},\frac{8}{3\sqrt{10}})$
• D. $(\frac{1}{3\sqrt{10}},\frac{-1}{3\sqrt{10}},\frac{1}{3\sqrt{10}})$
• E. $(\frac{1}{3\sqrt{10}},\frac{5}{3\sqrt{10}},\frac{8}{3\sqrt{10}})$

Hint:

Vector Tip: Direction cosines are simply the scalar components of the "unit vector" pointing in the exact same direction as the original vector!

Answer 1

The Correct Option is C

Concept:
The direction cosines of a vector $\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$ represent the cosines of the angles the vector makes with the coordinate axes. They are found by dividing each individual component of the vector by its total magnitude $|\vec{v}| = \sqrt{a^2 + b^2 + c^2}$.

Step 1: Identify the vector components.
From the given vector $\vec{v} = \hat{i} - 5\hat{j} + 8\hat{k}$, we extract the scalar components: $$a = 1$$ $$b = -5$$ $$c = 8$$

Step 2: Set up the magnitude calculation.
Use the 3D distance formula to find the magnitude $|\vec{v}|$: $$|\vec{v}| = \sqrt{1^2 + (-5)^2 + 8^2}$$

Step 3: Calculate the sum of squares.
Evaluate the squares inside the square root: $$|\vec{v}| = \sqrt{1 + 25 + 64}$$ $$|\vec{v}| = \sqrt{90}$$

Step 4: Simplify the radical expression.
Factor out the largest perfect square from 90 to simplify the radical: $$|\vec{v}| = \sqrt{9 \times 10} = 3\sqrt{10}$$

Step 5: Determine the direction cosines.
Divide each original component by the simplified magnitude to establish the direction cosines $(l, m, n)$: $$\left( \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|}, \frac{c}{|\vec{v}|} \right) = \left( \frac{1}{3\sqrt{10}}, \frac{-5}{3\sqrt{10}}, \frac{8}{3\sqrt{10}} \right)$$ Hence the correct answer is (C) $(\frac{1}{3\sqrt{10}},\frac{-5}{3\sqrt{10}},\frac{8}{3\sqrt{10}})$.