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. \(-\hat{i} + \hat{j} - 2\hat{k}\)

Hint:

To find a vector from one point to another, subtract the coordinates of the initial point from the corresponding coordinates of the terminal point: \[ \overrightarrow{BA} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k}. \]

Answer 1

The Correct Option is D

The vector from \( B(3, -4, 7) \) to \( A(2, -3, 5) \) is calculated as \( \overrightarrow{BA} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k} \), where \( (x_1, y_1, z_1) \) are the coordinates of \( B \) and \( (x_2, y_2, z_2) \) are the coordinates of \( A \).

Substituting the coordinates yields: \[ \overrightarrow{BA} = (2 - 3)\hat{i} + (-3 - (-4))\hat{j} + (5 - 7)\hat{k}. \] 

Simplifying each component: \[ \overrightarrow{BA} = (-1)\hat{i} + (1)\hat{j} + (-2)\hat{k}. \] 

Therefore, the vector is expressed as: \[ \overrightarrow{BA} = -\hat{i} + \hat{j} - 2\hat{k}. \] 

The resulting vector is \(-\hat{i} + \hat{j} - 2\hat{k}\), corresponding to option (D).