Question

Let \( \vec{a} = 3\vec{i} + \vec{j} - 2\vec{k} \), \( \vec{b} = -5\vec{i} + 7\vec{j} \), and \( \vec{c} = 3\vec{i} + y\vec{j} \) be three vectors such that \( |\vec{a} - \vec{b} + \vec{c}| = \sqrt{141} \). If \( y_1 \) and \( y_2 \) are the values of \( y \) satisfying the given condition, then \( |y_1 - y_2| = \)

• A. \( 12 \)
• B. \( 11 \)
• C. \( 9 \)
• D. \( 8 \)

Hint:

Use vector addition and magnitude identity to build and solve a quadratic equation.

Answer 1

The Correct Option is D

Compute \( \vec{a} - \vec{b} + \vec{c} \): \[ (3 + 5 + 3)\vec{i} + (1 - 7 + y)\vec{j} + (-2 + 0 + 0)\vec{k} = 11\vec{i} + (y - 6)\vec{j} - 2\vec{k} \] Magnitude: \[ |\vec{r}| = \sqrt{11^2 + (y - 6)^2 + (-2)^2} = \sqrt{121 + (y - 6)^2 + 4} = \sqrt{125 + (y - 6)^2} \] Set equal to given magnitude: \[ \sqrt{125 + (y - 6)^2} = \sqrt{141} \Rightarrow (y - 6)^2 = 16 \Rightarrow y - 6 = \pm 4 \] \[ y_1 = 10, y_2 = 2 \Rightarrow |y_1 - y_2| = 8 \]