Question

Let \(\vec{a}\), \(\vec{b}\) be position vectors of points \(A\) and \(B\) respectively. \(C\) and \(D\) are points on the line \(AB\) such that \(\overrightarrow{AB}, \overrightarrow{AC}\) and \(\overrightarrow{BD}, \overrightarrow{BA}\) are two pairs of like vectors. If \(\overrightarrow{AC} = 3 \overrightarrow{AB}\) and \(\overrightarrow{BD} = 2 \overrightarrow{BA}\), then \(\overrightarrow{CD} =\)

• A. \(3\vec{b} - 4 \vec{a}\)
• B. \(4 \vec{a} - 4 \vec{b}\)
• C. \(4 \vec{a} - 3 \vec{b}\)
• D. \(3 \vec{b} - 3 \vec{a}\)

Hint:

Use vector addition and subtraction carefully to find vectors between points.

Answer 1

The Correct Option is B

Step 1: Express vectors
\[ \overrightarrow{AB} = \vec{b} - \vec{a} \] Step 2: Given
\[ \overrightarrow{AC} = 3 \overrightarrow{AB} = 3(\vec{b} - \vec{a}) = 3\vec{b} - 3\vec{a} \] \[ \overrightarrow{BD} = 2 \overrightarrow{BA} = 2(\vec{a} - \vec{b}) = 2 \vec{a} - 2 \vec{b} \] Step 3: Find \(\overrightarrow{CD}\)
\[ \overrightarrow{CD} = \overrightarrow{BD} - \overrightarrow{BC} = \overrightarrow{BD} - (\vec{c} - \vec{b}) = \overrightarrow{BD} - (\overrightarrow{AC}) = (2 \vec{a} - 2 \vec{b}) - (3 \vec{b} - 3 \vec{a}) = 2 \vec{a} - 2 \vec{b} - 3 \vec{b} + 3 \vec{a} = 5 \vec{a} - 5 \vec{b} \] But since \(\overrightarrow{AC} = 3 \overrightarrow{AB}\), \(\vec{c} = \vec{a} + 3 (\vec{b} - \vec{a}) = 3 \vec{b} - 2 \vec{a}\). Then, \[ \overrightarrow{BC} = \vec{c} - \vec{b} = (3 \vec{b} - 2 \vec{a}) - \vec{b} = 2 \vec{b} - 2 \vec{a} \] \[ \overrightarrow{CD} = \overrightarrow{BD} - \overrightarrow{BC} = (2 \vec{a} - 2 \vec{b}) - (2 \vec{b} - 2 \vec{a}) = 4 \vec{a} - 4 \vec{b} \]