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Vector Algebra question

2019 · 10 Apr · Shift 2 · Q30
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Vector Algebra question

2019 · 10 Apr · Shift 2 · Q30

JEE MainMathematicsVector AlgebraMCQ+4 / −1
The distance of the point having position vector −i^+2j^+6k^- \widehat i + 2\widehat j + 6\widehat k−i+2j​+6k from the straight line passing through the point (2, 3, – 4) and parallel to the vector, 6i^+3j^−4k^6\widehat i + 3\widehat j - 4\widehat k6i+3j​−4k is :
  1. A
    6
  2. B
    7
  3. C
    2132\sqrt {13}213​
  4. D
    434\sqrt 343​
View written solutionFree

Correct answer: B

  1. Identify the point and the line

The given point has position vector −i^+2j^+6k^,-\hat i+2\hat j+6\hat k,−i^+2j^​+6k^, so its coordinates are P(−1,2,6).P(-1,2,6).P(−1,2,6).

The line passes through A(2,3,−4)A(2,3,-4)A(2,3,−4) and is parallel to the vector d⃗=6i^+3j^−4k^=(6,3,−4).\vec d=6\hat i+3\hat j-4\hat k=(6,3,-4).d=6i^+3j^​−4k^=(6,3,−4).

  1. Use the formula for distance of a point from a line in 3D

If a line passes through point AAA with direction vector d⃗\vec dd, and PPP is the given point, then the distance is Distance=∣AP→×d⃗∣∣d⃗∣.\text{Distance} = \frac{|\overrightarrow{AP}\times \vec d|}{|\vec d|}.Distance=∣d∣∣AP×d∣​.

Here, AP→=P−A=(−1−2,  2−3,  6−(−4))=(−3,−1,10).\overrightarrow{AP}=P-A=(-1-2,\;2-3,\;6-(-4))=(-3,-1,10).AP=P−A=(−1−2,2−3,6−(−4))=(−3,−1,10).

  1. Compute the cross product
\begin{vmatrix} \hat i & \hat j & \hat k \\ -3 & -1 & 10 \\ 6 & 3 & -4 \end{vmatrix}$$ $$=\hat i\big((-1)(-4)-10\cdot 3\big)-\hat j\big((-3)(-4)-10\cdot 6\big)+\hat k\big((-3)(3)-(-1)(6)\big).$$ Now simplify: - Coefficient of $\hat i$: $$4-30=-26$$ - Coefficient of $\hat j$: $$-(12-60)=48$$ - Coefficient of $\hat k$: $$-9+6=-3$$ So, $$\overrightarrow{AP}\times \vec d=(-26,48,-3).$$ --- 4. **Find its magnitude** $$|\overrightarrow{AP}\times \vec d|=\sqrt{(-26)^2+48^2+(-3)^2}$$ $$=\sqrt{676+2304+9}$$ $$=\sqrt{2989}.$$ Also, $$|\vec d|=\sqrt{6^2+3^2+(-4)^2}= \sqrt{36+9+16}= \sqrt{61}.$$ Thus distance is $$\frac{\sqrt{2989}}{\sqrt{61}}= \sqrt{\frac{2989}{61}}= \sqrt{49}=7.$$ --- 5. **Match with options** The distance is $$7.$$ So the correct option is **B**. --- 6. **Comparison with stored answer** Stored correct answer: **B** Our derived answer: **B** Hence, the derived answer agrees with the stored answer.
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