Question

In a trapezium \(ABCD\), \[ \overrightarrow{BC}=\lambda \overrightarrow{AD} \] and \[ \vec{x}=\overrightarrow{AC}+\overrightarrow{BD}. \] If \[ \vec{x}=p\overrightarrow{AD}, \] then \(p=\)

• A. \(\lambda-1\)
• B. \(\lambda+1\)
• C. \(1-\lambda\)
• D. \(2\lambda-1\)

Hint:

In vector problems, first express all vectors using a common set of vectors and then simplify by cancellation.

Answer 1

The Correct Option is B

Step 1: Express AC in terms of AB and BC.
Using the triangle law of vector addition in triangle $ABC$: \[ \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}. \] We are given $\overrightarrow{BC} = \lambda\,\overrightarrow{AD}$, so: \[ \overrightarrow{AC} = \overrightarrow{AB} + \lambda\,\overrightarrow{AD}. \]
Step 2: Express BD in terms of BA and AD.
Using triangle law in triangle $ABD$: \[ \overrightarrow{BD} = \overrightarrow{BA} + \overrightarrow{AD} = -\overrightarrow{AB} + \overrightarrow{AD}. \]
Step 3: Add the two diagonal vectors.
\[ \vec{x} = \overrightarrow{AC}+\overrightarrow{BD} = (\overrightarrow{AB}+\lambda\,\overrightarrow{AD})+(-\overrightarrow{AB}+\overrightarrow{AD}). \]
Step 4: Collect like terms.
The $\overrightarrow{AB}$ terms cancel: \[ \vec{x} = (\lambda+1)\overrightarrow{AD}. \]
Step 5: Compare with the given form.
We are told $\vec{x} = p\,\overrightarrow{AD}$. Comparing, $p = \lambda+1$.
Step 6: State the final answer.
\[ \boxed{p = \lambda+1} \]