For any vector a=a_1 i+a_2 j+a_3 k, with 10 |a_i |<1, i=1,2,3, co… - Vidyadip

MCQ

For any vector $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$, with $10\left|a_i\right|<1, i=1,2,3$, consider the following statements:

$(A)$: $\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\} \leq|\vec{a}|$

$(B)$: $|\vec{a}| \leq 3 \max \left\{\left| a _1\right|,\left| a _2\right|,\left| a _3\right|\right\}$

A
Only $(B)$ is true
B
Only $(A)$ is true
C
Neither $(A)$ nor $(B)$ is true
✓
Both $(A)$ and $(B)$ are true

✓

Answer

Correct option: D.

Both $(A)$ and $(B)$ are true

d
Without loss of generality

Let $\left|a_1\right| \leq\left|a_2\right| \leq\left|a_3\right|$

$|\vec{a}|^2=\left|a_1\right|^2+\left|a_2\right|^2+\left|a_3\right|^2 \geq\left(a_3\right)^2$

$|\vec{a}| \geq\left|a_3\right|=\max \left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}$

$A$ is true

$|\vec{a}|^2=\left| a _1\right|^2+\left| a _2\right|^2+\left| a _3\right|^2 \leq\left| a _3\right|^2+\left| a _3\right|^2+\left| a _3\right|^2$

$|\overrightarrow{ a }|^2 \leq 3\left| a _3\right|^2$

$|\overrightarrow{ a }| \leq \sqrt{3}\left| a _3\right|=\sqrt{3} \max \left\{\left| a _1\right|,\left| a _2\right|,\left| a _3\right|\right\}$

$\leq 3 \max \left\{\left| a _1\right|,\left| a _2\right|,\left| a _3\right|\right\}$

$(2)$ is true

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