The projections of a vector on the three coordinate axis are 6,-3,2 respectively. The direction cosines of the vector are :

6 \over 7 , - 3 \over 7,2 \over 7

6 \over 5, - 3 \over 5,2 \over 5

- 6 \over 7 , - 3 \over 7,2 \over 7

3D Geometry PYQ — JEE Main 2009 · Shift 0 · Q46 with Solution

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Correct answer: B

The projections of the vector on the three coordinate axes are given as:

-3, 2$$ So the vector can be written as $$\vec{v} = 6\hat{i} - 3\hat{j} + 2\hat{k}.$$ 2. **Find the magnitude of the vector** The magnitude is $$|\vec{v}| = \sqrt{6^2 + (-3)^2 + 2^2} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7.$$ 3. **Use the formula for direction cosines** If the direction cosines are $(l,m,n)$, then $$l = \frac{x}{|\vec{v}|}, \quad m = \frac{y}{|\vec{v}|}, \quad n = \frac{z}{|\vec{v}|}.$$ Hence, $$l = \frac{6}{7}, \quad m = \frac{-3}{7}, \quad n = \frac{2}{7}.$$ So the direction cosines are $$\left(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right).$$ 4. **Check the options** - **A:** $\left(\frac{6}{5}, -\frac{3}{5}, \frac{2}{5}\right)$ — incorrect, direction cosines must satisfy $l^2+m^2+n^2=1$. - **B:** $\left(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right)$ — correct. - **C:** $\left(-\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right)$ — incorrect sign for first cosine. - **D:** $(6,-3,2)$ — incorrect, these are components, not direction cosines. Therefore, the correct option is **B**.

View written solutionFree

Correct answer: B

The projections of the vector on the three coordinate axes are given as:

-3, 2$$ So the vector can be written as $$\vec{v} = 6\hat{i} - 3\hat{j} + 2\hat{k}.$$ 2. **Find the magnitude of the vector** The magnitude is $$|\vec{v}| = \sqrt{6^2 + (-3)^2 + 2^2} = \sqrt{36 + 9 + 4} = \sqrt{49} = 7.$$ 3. **Use the formula for direction cosines** If the direction cosines are $(l,m,n)$, then $$l = \frac{x}{|\vec{v}|}, \quad m = \frac{y}{|\vec{v}|}, \quad n = \frac{z}{|\vec{v}|}.$$ Hence, $$l = \frac{6}{7}, \quad m = \frac{-3}{7}, \quad n = \frac{2}{7}.$$ So the direction cosines are $$\left(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right).$$ 4. **Check the options** - **A:** $\left(\frac{6}{5}, -\frac{3}{5}, \frac{2}{5}\right)$ — incorrect, direction cosines must satisfy $l^2+m^2+n^2=1$. - **B:** $\left(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right)$ — correct. - **C:** $\left(-\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}\right)$ — incorrect sign for first cosine. - **D:** $(6,-3,2)$ — incorrect, these are components, not direction cosines. Therefore, the correct option is **B**.