Question:
Three vectors of magnitudes \(a\), \(2a\), \(3a\) are along the directions of the diagonals of \(3\) adjacent faces of a cube that meet in a point. Then the magnitude of the sum of these vectors is
Three vectors of magnitudes \(a\), \(2a\), \(3a\) are along the directions of the diagonals of \(3\) adjacent faces of a cube that meet in a point. Then the magnitude of the sum of these vectors is
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The diagonal of a face of a cube lies equally along two perpendicular edge directions. Represent face diagonals using unit vectors before adding the vectors.
Updated On: Jun 22, 2026
- \(4a\)
- \(5a\)
- \(6a\)
- \(8a\)
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The Correct Option is B
Solution and Explanation
Step 1: Choose coordinate axes along the edges of the cube.
Let the three mutually perpendicular edges of the cube meeting at a point be along: \[ \hat i,\quad \hat j,\quad \hat k \] The diagonals of the three adjacent faces through this point have directions: \[ \hat i+\hat j, \] \[ \hat j+\hat k, \] \[ \hat k+\hat i \] Their unit vectors are: \[ \frac{\hat i+\hat j}{\sqrt2}, \qquad \frac{\hat j+\hat k}{\sqrt2}, \qquad \frac{\hat k+\hat i}{\sqrt2} \]
Step 2: Write the three vectors.
The vectors have magnitudes \[ a,\quad 2a,\quad 3a \] Hence, \[ \vec A= a\cdot \frac{\hat i+\hat j}{\sqrt2} \] \[ \vec B= 2a\cdot \frac{\hat j+\hat k}{\sqrt2} \] \[ \vec C= 3a\cdot \frac{\hat k+\hat i}{\sqrt2} \]
Step 3: Find the resultant vector.
\[ \vec R=\vec A+\vec B+\vec C \] Substituting, \[ \vec R= \frac{a}{\sqrt2}(\hat i+\hat j) + \frac{2a}{\sqrt2}(\hat j+\hat k) + \frac{3a}{\sqrt2}(\hat k+\hat i) \] Collecting coefficients, \[ \vec R= \frac{a}{\sqrt2} [ (1+3)\hat i+(1+2)\hat j+(2+3)\hat k ] \] \[ \vec R= \frac{a}{\sqrt2} (4\hat i+3\hat j+5\hat k) \]
Step 4: Find the magnitude of \(\vec R\).
\[ |\vec R| = \frac{a}{\sqrt2} \sqrt{4^2+3^2+5^2} \] \[ = \frac{a}{\sqrt2} \sqrt{16+9+25} \] \[ = \frac{a}{\sqrt2} \sqrt{50} \] \[ = \frac{a}{\sqrt2}\cdot 5\sqrt2 \] \[ =5a \]
Step 5: Final conclusion.
Therefore, the magnitude of the sum is \[ \boxed{5a} \]
Let the three mutually perpendicular edges of the cube meeting at a point be along: \[ \hat i,\quad \hat j,\quad \hat k \] The diagonals of the three adjacent faces through this point have directions: \[ \hat i+\hat j, \] \[ \hat j+\hat k, \] \[ \hat k+\hat i \] Their unit vectors are: \[ \frac{\hat i+\hat j}{\sqrt2}, \qquad \frac{\hat j+\hat k}{\sqrt2}, \qquad \frac{\hat k+\hat i}{\sqrt2} \]
Step 2: Write the three vectors.
The vectors have magnitudes \[ a,\quad 2a,\quad 3a \] Hence, \[ \vec A= a\cdot \frac{\hat i+\hat j}{\sqrt2} \] \[ \vec B= 2a\cdot \frac{\hat j+\hat k}{\sqrt2} \] \[ \vec C= 3a\cdot \frac{\hat k+\hat i}{\sqrt2} \]
Step 3: Find the resultant vector.
\[ \vec R=\vec A+\vec B+\vec C \] Substituting, \[ \vec R= \frac{a}{\sqrt2}(\hat i+\hat j) + \frac{2a}{\sqrt2}(\hat j+\hat k) + \frac{3a}{\sqrt2}(\hat k+\hat i) \] Collecting coefficients, \[ \vec R= \frac{a}{\sqrt2} [ (1+3)\hat i+(1+2)\hat j+(2+3)\hat k ] \] \[ \vec R= \frac{a}{\sqrt2} (4\hat i+3\hat j+5\hat k) \]
Step 4: Find the magnitude of \(\vec R\).
\[ |\vec R| = \frac{a}{\sqrt2} \sqrt{4^2+3^2+5^2} \] \[ = \frac{a}{\sqrt2} \sqrt{16+9+25} \] \[ = \frac{a}{\sqrt2} \sqrt{50} \] \[ = \frac{a}{\sqrt2}\cdot 5\sqrt2 \] \[ =5a \]
Step 5: Final conclusion.
Therefore, the magnitude of the sum is \[ \boxed{5a} \]
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