Statement I: If , and and be two nonzero vectors such that and , then for all Statement II: If the vectors and represent the sides of regular hexagon, then Statement III: Four points A,B,C and D with position vectors and respectively, are c

Question

Which of the following must be the true value of the statements below in that order.

Statement I: If $\vec{a} = 2 \hat{i} + \hat{j} + \hat{k}$ , and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $| \vec{a} + \vec{b} + \vec{c} |$ $=$ $| \vec{a} + \vec{b} - \vec{c} |$ and $\vec{b} \cdot \vec{c} = 0$ , then $| \vec{a} + λ \vec{c} |$ $\geq$ $| \vec{a} |$ for all $λ \in R$

Statement II: If the vectors $\vec{P Q}, \vec{Q R}, \vec{R S}, \vec{S T}, \vec{T U}$ and $\vec{U P}$ represent the sides of regular hexagon, then $\vec{P Q} \times (\vec{R S} + \vec{S T}) \neq \vec{0}$

Statement III: Four points A,B,C and D with position vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ respectively, are coplanar, then there exist constants $x, y, z$ and $w$ such that $x \vec{a} + y \vec{b} + z \vec{c} + w \vec{d} = \vec{0}$ where $x + y + z + w = 0$ but not all $x, y, z$ and $w$ are zero.

Smart Achievers · Accepted Answer

(4)

$Since, given$ $| a + b + c |$ $=$ $| a + b - c |$

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

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

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

$So, (B) is incorrect$

$Now,$ $| \vec{a} + λ \vec{c} |^{2}$ $\geq$ $| \vec{a} |^{2}$

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

$\vec{d}$ $(using triangle law)$

$=$ $| \vec{P Q} |$ $\times$ $| \vec{R T} |$ $\sin 150 ° \neq 0 \Rightarrow Statement-1 is true.$

$Also, \vec{P Q} \times \vec{R S} =$ $| \vec{P Q} |$ $\times$ $| \vec{R S} |$ $\sin 120 ° \times {\hat{n}}_{1} \neq 0$

$And \vec{P Q} \times \vec{S T} =$ $| \vec{P Q} |$ $\times$ $| \vec{S T} |$ $\sin 180 ° \times {\hat{n}}_{2} \neq 0$

$∴$ $Statement-2 is false$

Solution

1 TFF

2 TFT

3 FFT

4 TTT

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