The two vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}+5\hat{k}$ represent the two sides $\vec{AB}$ and $\vec{AC}$ respectively of a ΔABC. The length of the median through A is

Question

The two vectors $\hat{i}+\hat{j}+\hat{k}$  and  $\hat{i}+3\hat{j}+5\hat{k}$  represent the two sides  $\vec{AB}$  and  $\vec{AC}$  respectively of a ΔABC. The length of the median through A is

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The two vectors $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + 3\hat{j} + 5\hat{k}$ represent the two sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ respectively of a $\triangle ABC$. We need to find the length of the median through A. Let D be the midpoint of BC. The median through A is AD. We can find $\overrightarrow{AD}$ using the formula for the midpoint: $\overrightarrow{AD} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$ $\overrightarrow{AD} = \frac{1}{2} ((\hat{i} + \hat{j} + \hat{k}) + (\hat{i} + 3\hat{j} + 5\hat{k}))$ $\overrightarrow{AD} = \frac{1}{2} (2\hat{i} + 4\hat{j} + 6\hat{k}) = \hat{i} + 2\hat{j} + 3\hat{k}$ The length of the median is the magnitude of $\overrightarrow{AD}$: $|\overrightarrow{AD}| = \sqrt{(1)^2 + (2)^2 + (3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$ Therefore, the length of the median through A is $\sqrt{14}$. Thus, the correct option is (D) $\sqrt{14}$ .

Answer

$\frac{\sqrt{14}}{2}$

Answer

Answer

Answer

$\sqrt{14}$

List-I	List-II
(A) 4î − 2ĵ − 4k̂	(I) A vector perpendicular to both î + 2ĵ + k̂ and 2î + 2ĵ + 3k̂
(B) 4î − 4ĵ + 2k̂	(II) Direction ratios are −2, 1, 2
(C) 2î − 4ĵ + 4k̂	(III) Angle with the vector î − 2ĵ − k̂ is cos⁻¹(1/√6)
(D) 4î − ĵ − 2k̂	(IV) Dot product with −2î + ĵ + 3k̂ is 10

The two vectors \(\hat{i}+\hat{j}+\hat{k}\) and \(\hat{i}+3\hat{j}+5\hat{k}\) represent the two sides \(\vec{AB}\) and \(\vec{AC}\) respectively of a ΔABC. The length of the median through A is

The Correct Option is D

Solution and Explanation

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