Question:
The position vectors of the vertices \(A\) and \(B\) of a triangle \(ABC\) are
\[
\hat{i}+3\hat{j}+4\hat{k}
\]
and
\[
2\hat{i}+\hat{j}+2\hat{k}
\]
respectively. If
\[
|AC|=5
\]
and
\[
\angle A=\frac{\pi}{3},
\]
then
\[
|BC|=
\]
The position vectors of the vertices \(A\) and \(B\) of a triangle \(ABC\) are
\[
\hat{i}+3\hat{j}+4\hat{k}
\]
and
\[
2\hat{i}+\hat{j}+2\hat{k}
\]
respectively. If
\[
|AC|=5
\]
and
\[
\angle A=\frac{\pi}{3},
\]
then
\[
|BC|=
\]
Show Hint
When coordinates of two vertices and an included angle are given, first find the known side using the distance formula and then use the cosine rule to determine the required side.
Updated On: Jul 29, 2026
- \(\sqrt{26}\)
- \(3\sqrt{19}\)
- \(3\sqrt{26}\)
- \(\sqrt{19}\)
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
First find the length of side \(AB\) using the distance formula. Then apply the cosine rule in \(\triangle ABC\).
Step 1: Find the length of \(AB\). The position vectors are \[ A(1,3,4), \qquad B(2,1,2). \] Therefore, \[ AB = \sqrt{(2-1)^2+(1-3)^2+(2-4)^2}. \] \[ = \sqrt{1+4+4}. \] \[ = 3. \] Thus, \[ |AB|=3. \]
Step 2: Apply the cosine rule. Given, \[ |AC|=5, \qquad \angle A=\frac{\pi}{3}. \] Using the cosine rule, \[ BC^2 = AB^2+AC^2 - 2(AB)(AC)\cos A. \] \[ = 3^2+5^2 - 2(3)(5)\cos\frac{\pi}{3}. \] \[ = 9+25 - 30\left(\frac12\right). \] \[ = 34-15. \] \[ = 19. \]
Step 3: Find \(BC\). \[ BC=\sqrt{19}. \] Therefore, \[ \boxed{|BC|=\sqrt{19}} \] \[ \boxed{\text{Answer = (D)}} \]
Step 1: Find the length of \(AB\). The position vectors are \[ A(1,3,4), \qquad B(2,1,2). \] Therefore, \[ AB = \sqrt{(2-1)^2+(1-3)^2+(2-4)^2}. \] \[ = \sqrt{1+4+4}. \] \[ = 3. \] Thus, \[ |AB|=3. \]
Step 2: Apply the cosine rule. Given, \[ |AC|=5, \qquad \angle A=\frac{\pi}{3}. \] Using the cosine rule, \[ BC^2 = AB^2+AC^2 - 2(AB)(AC)\cos A. \] \[ = 3^2+5^2 - 2(3)(5)\cos\frac{\pi}{3}. \] \[ = 9+25 - 30\left(\frac12\right). \] \[ = 34-15. \] \[ = 19. \]
Step 3: Find \(BC\). \[ BC=\sqrt{19}. \] Therefore, \[ \boxed{|BC|=\sqrt{19}} \] \[ \boxed{\text{Answer = (D)}} \]
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