Q

The value of $\dpi{100} a$, for which the points $\dpi{100} A,B,C$ with position  vectors $\dpi{100} 2\hat{i}-\hat{j}+\hat{k},\; \; \hat{i}-3\hat{j}-5\hat{k}\; \; and\; \; a\hat{i}-3\hat{j}+\hat{k}$  respectively are the vertices of a right angled triangle at $\dpi{100} c$ are

• Option 1)

2 and 1

• Option 2)

–2 and –1

• Option 3)

–2 and 1

• Option 4)

2 and –1

G gaurav

As we have learned

Scalar Product of two vectors -

$\vec{a}.\vec{b}> 0 \:an\: acute\: angle$

$\vec{a}.\vec{b}< 0 \:an\: obtuse\: angle$

$\vec{a}.\vec{b}= 0 \:a\:right\: angle$

- wherein

$\Theta$  is the angle between the vectors $\vec{a}\:and\:\vec{b}$

Position Vector -

If $\vec{a}$ and $\vec{b}$ are the position of vectors of two points A and B then

$\overrightarrow{AB}= \vec{b}-\vec{a}$

$\overrightarrow{AB}= P \vee of B - P\vee of A$

- wherein

$\vec{AC}\cdot \vec{BC} = 0$

$\Rightarrow ((a\hat{i}-3\hat{j}+\hat{k})-(2\hat{i}-\hat{j}+\hat{k}))\cdot ((a\hat{i}-3\hat{j}+\hat{k})- (\hat{i}-3\hat{j}-5\hat{k}))=0$

( a-2 )( a-1 )= 0

a = 1,2

Option 1)

2 and 1

Option 2)

–2 and –1

Option 3)

–2 and 1

Option 4)

2 and –1

Let $\dpi{100} a,b \; and\, \; c$ be distinct non­-negative numbers. If the vectors $\dpi{100} a\hat{i}+a\hat{j}+c\hat{k},\; \hat{i}+\hat{k}\; and\; c\hat{i}+c\hat{j}+b\hat{k}$  lie in a plane, then $\dpi{100} c$ is

• Option 1)

the arithmetic mean of $a\; \; and\; \; b$

• Option 2)

the geometric mean of  $a\; \; and\; \; b$

• Option 3)

the harmonic mean of $a\; \; and\; \; b$

• Option 4)

equal to zero

G gaurav

As we have learned

Scalar Triple Product -

$\left [ \vec{a}\;\vec{b}\; \vec{c} \right ]$

$=\left (\vec{a}\times \vec{b}\right)\cdot \vec{c}= \vec{a}\cdot \left ( \vec{b} \times \vec{c}\right )$

$=\left (\vec{b}\times \vec{c}\right)\cdot \vec{a}= \vec{b}\cdot \left ( \vec{c} \times \vec{a}\right )$

$=\left (\vec{c}\times \vec{a}\right)\cdot \vec{b}= \vec{c}\cdot \left ( \vec{a} \times \vec{b}\right )$

- wherein

Scalar Triple Product of three vectors $\hat{a},\hat{b},\hat{c}$.

$[ a\hat{i}+a\hat{j}+c\hat{k}\; \;\; \; \; \; \hat{i}+\hat{k}\; \; \; \; \; \; c\hat{i}+c\hat{j}+b\hat{k}] = 0$

$\begin{vmatrix} a & a &c \\ 1& 0 & 1\\ c&c & b \end{vmatrix} = 0$

$- ac -a (b-c)+c^2 = 0 \\ ab = c^2$

Option 1)

the arithmetic mean of $a\; \; and\; \; b$

Option 2)

the geometric mean of  $a\; \; and\; \; b$

Option 3)

the harmonic mean of $a\; \; and\; \; b$

Option 4)

equal to zero

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