If A B = C + D, then select the correct alternative- - Vidyadip

MCQ

If $\overrightarrow A \times \overrightarrow B = \overrightarrow C + \overrightarrow D,$ then select the correct alternative-

A
$\overrightarrow B$ is parallel to $\overrightarrow C + \overrightarrow D$
B
$\overrightarrow A$ is perpendicular to $\overrightarrow C$
C
Component of $\overrightarrow C$ along $\overrightarrow A = $ component of $\overrightarrow D$ along $\overrightarrow A$
✓
Component of $\overrightarrow C$ along $\overrightarrow A = -$ component of $\overrightarrow D $ along $\overrightarrow A$

✓

Answer

Correct option: D.

Component of $\overrightarrow C$ along $\overrightarrow A = -$ component of $\overrightarrow D $ along $\overrightarrow A$

d
According to definition of $cross-product$

$\overrightarrow{\mathrm{C}}+\overrightarrow{\mathrm{D}}$ is perpendicular to both $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$

i.e. $\quad \vec{A} \cdot(\vec{C}+\vec{D})=0$

or $\quad \overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{C}}+\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{D}}=0$

or $\quad$ A (component of $\overrightarrow{\mathrm{C}}$ along $\overrightarrow{\mathrm{A}}$ ) $+\mathrm{A}$

(component of $\overrightarrow{\mathrm{D}} \text { along } \overrightarrow{\mathrm{A}})=0$

or component of $\overrightarrow{\mathrm{C}}$ along $\overrightarrow{\mathrm{A}}$

$=$ $-$ component of $\overrightarrow{\mathrm{D}}$ along $\overrightarrow{\mathrm{A}}$

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