Question

The vector with terminal point \( A(2, -3, 5) \) and initial point \( B(3, -4, 7) \) is:

• A. \( \hat{i} - \hat{j} + 2\hat{k} \)
• B. \( \hat{i} + \hat{j} + 2\hat{k} \)
• C. \( -\hat{i} - \hat{j} - 2\hat{k} \)
• D. \( -\hat{i} + \hat{j} - 2\hat{k} \)
• E.

Hint:

To find the vector between two points, subtract the coordinates of the initial point from the corresponding coordinates of the terminal point: \[ \vec{v} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}. \]

Answer 1

The Correct Option is D

The vector from the initial point \( B(3, -4, 7) \) to the terminal point \( A(2, -3, 5) \) is given by: \[ \vec{v} = \vec{A} - \vec{B}. \] Calculate the components of the vector: \[ \vec{v} = (2 - 3)\hat{i} + (-3 - (-4))\hat{j} + (5 - 7)\hat{k}. \] Simplify each component: \[ \vec{v} = (-1)\hat{i} + (1)\hat{j} + (-2)\hat{k}. \] Thus: \[ \vec{v} = -\hat{i} + \hat{j} - 2\hat{k}. \] Hence, the vector is \( -\hat{i} + \hat{j} - 2\hat{k} \), and the correct answer is (D).