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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
MCQ
Let $\overrightarrow{ a }=2 \hat{ i }+\hat{ j }+\hat{ k }$, and $\overrightarrow{ b }$ and $\overrightarrow{ c }$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}| \quad$ and $\vec{b} \cdot \vec{c}=0$. Consider the following two statement:

$(A)$ $|\overrightarrow{ a }+\lambda \overrightarrow{ c }| \geq|\overrightarrow{ a }|$ for all $\lambda \in R$.

$(B)$ $\overrightarrow{ a }$ and $\overrightarrow{ c }$ are always parallel

  • A
    only $(B)$ is correct
  • B
    neither $(A)$ nor $(B)$ is correct
  • ✓
    only $(A)$ is correct
  • D
    both $(A)$ and $(B)$ are correct.
✓

Answer

Correct option: C.
only $(A)$ is correct
c
$|\vec{a}+\vec{b}+\vec{c}|^2=|\vec{a}+\vec{b}-\vec{c}|^2$

$2 \vec{a} \cdot \vec{b}+2 \vec{b} \cdot \vec{c}+2 \vec{c} \cdot \vec{a}=2 \vec{a} \cdot \vec{b}-2 \vec{b} \cdot \vec{c}-2 \vec{c} \cdot \vec{a}$

$4 \vec{a} \cdot \vec{c}=0$

$B$ is incorrect

$|\overrightarrow{ a }+\lambda \overrightarrow{ c }|^2 \geq|\overrightarrow{ a }|^2$

$\lambda^2 c ^2 \geq 0$

True $\forall \lambda \in R$ $(A)$ is correct.

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