Question:
The image of the point with position vector \( \hat{i} + 3\hat{k} \) in the plane \( \vec{r}\cdot(\hat{i} + \hat{j} + \hat{k}) = 1 \) is
The image of the point with position vector \( \hat{i} + 3\hat{k} \) in the plane \( \vec{r}\cdot(\hat{i} + \hat{j} + \hat{k}) = 1 \) is
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Use reflection formula to find image of a point in a plane quickly.
Updated On: Apr 23, 2026
- \( \hat{i} + 2\hat{j} + \hat{k} \)
- \( \hat{i} - 2\hat{j} + \hat{k} \)
- \( -\hat{i} - 2\hat{j} + \hat{k} \)
- \( \hat{i} + 2\hat{j} - \hat{k} \)
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The Correct Option is B
Solution and Explanation
Concept:
Image of a point in a plane is found using reflection formula.
Step 1: Given point: \[ P = (1,0,3) \] Plane: \[ x + y + z = 1 \]
Step 2: Use reflection formula: \[ \text{Image} = P - 2\frac{ax_1 + by_1 + cz_1 - d}{a^2+b^2+c^2}(a,b,c) \] Substitute: \[ = (1,0,3) - 2\frac{1+0+3-1}{3}(1,1,1) \] \[ = (1,0,3) - 2(1,1,1) = (1,-2,1) \] Final Answer: \[ \hat{i} - 2\hat{j} + \hat{k} \]
Step 1: Given point: \[ P = (1,0,3) \] Plane: \[ x + y + z = 1 \]
Step 2: Use reflection formula: \[ \text{Image} = P - 2\frac{ax_1 + by_1 + cz_1 - d}{a^2+b^2+c^2}(a,b,c) \] Substitute: \[ = (1,0,3) - 2\frac{1+0+3-1}{3}(1,1,1) \] \[ = (1,0,3) - 2(1,1,1) = (1,-2,1) \] Final Answer: \[ \hat{i} - 2\hat{j} + \hat{k} \]
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