Given three arbitrary vectors a, b, c, the vectors =5 a +6 b+7 c … - Vidyadip

MCQ

Given three arbitrary vectors $\bar{a}, \bar{b}, \bar{c}$, the vectors $\bar{\alpha}=5 \overline{ a }+6 \overline{b}+7 \overline{ c }, \beta=7 \overline{ a }-8 \overline{b}+9 \overline{ c }$, $\bar{\gamma}=3 \overline{ a }+20 \overline{b}+5 \overline{ c }$ are

A
Collinear
✓
Coplanar
C
Non-coplanar
D
None of these

✓

Answer

Correct option: B.

Coplanar

(B) $[\bar{\alpha} \bar{\beta} \bar{\gamma}]=\left|\begin{array}{ccc}5 & 6 & 7 \\ 7 & -8 & 9 \\ 3 & 20 & 5\end{array}\right|$
$=5(-40-180)-6(35-27)$ $+7(140+24)=0$
∴ the given vectors are coplanar.

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