Let a, b, c be three non-coplanar vectors and p, q, r defined by the relations p=(b × c)/([a ~b c]), q=(c × a)/([a ~b c]), r=(a × b)/([a ~b c]) then the value of the expression (a+b) · p+(b+c) · q+(c+a) · r is equal to

Correct answer: (D) 3. p ·(a+b) =p · a+p · b =((b × c) · a)/([a b c])+((b × c) · b)/([a b c]) =([b c a])/([a b c])+([b c b])/([a b c]) =1+0=1 Similarly, q ·(b+c)=1 and r ·(a+c)=1 (a+b) · p+(stackrel harpoonupb+c) · q+(c+a) · r =1+1+1 =3

Q98 / 150 · MHT CET 2024 (03 May Shift 1) · Mathematics

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ be three non-coplanar vectors and $\overline{\mathrm{p}}, \overline{\mathrm{q}}, \overline{\mathrm{r}}$ defined by the relations $\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}$ then the value of the expression $(\bar{a}+\bar{b}) \cdot \bar{p}+(\bar{b}+\bar{c}) \cdot \bar{q}+(\bar{c}+\bar{a}) \cdot \bar{r}$ is equal to

A
0
B
1
C
2
Correct answer
D
3

Solution

$\begin{aligned} \overline{\mathrm{p}} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}}) & =\overline{\mathrm{p}} \cdot \overline{\mathrm{a}}+\overline{\mathrm{p}} \cdot \overline{\mathrm{b}} \\ & =\frac{(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \cdot \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}+\frac{(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \cdot \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]} \\ & =\frac{[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{a}}]}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}+\frac{[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{b}}]}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]} \\ & =1+0=1\end{aligned}$ Similarly, $\overline{\mathrm{q}} \cdot(\overline{\mathrm{b}}+\overline{\mathrm{c}})=1$ and $\overline{\mathrm{r}} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{c}})=1$ $\begin{aligned} & (\overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot \overline{\mathrm{p}}+(\stackrel{\rightharpoonup}{\mathrm{b}}+\overline{\mathrm{c}}) \cdot \overline{\mathrm{q}}+(\overline{\mathrm{c}}+\overline{\mathrm{a}}) \cdot \overline{\mathrm{r}} \\ & =1+1+1 \\ & =3 \end{aligned}$

Hence, the correct answer is option: 'D'.

Solution

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