First slide

Cross product of vectors or vector product of vectors

Question

$If a^{2} + b^{2} + c^{2} = 1 where a, b, c \in R, then the maximum value of (4 a - 3 b)^{2} + (5 b - 4 c)^{2} + (3 c - 5 a)^{2} is$

Difficult

A

25
B

50
C

144
D

None of these

Solution

$\begin{array}{l} Sol. (b) Let {\vec{r}}_{1} = a \hat{i} + b \hat{j} + c \hat{k}, {\vec{r}}_{2} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \\ {|{\vec{r}}_{1} \times {\vec{r}}_{2}|}^{2} \leq {|{\vec{r}}_{1}|}^{2} {|{\vec{r}}_{2}|}^{2} - - - - - - - - (1) \\ Now {\vec{r}}_{1} \times {\vec{r}}_{2} = |\begin{array}{lll} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ 3 & 4 & 5 \end{array}| \end{array}$
$= \hat{i} (5 b - 4 c) + \hat{j} (3 c - 5 a) + \hat{k} (4 a - 3 b)$
$So, from (1)$
$(5 b - 4 c)^{2} + (3 c - 5 a)^{2} + (4 a - 3 b)^{2} \leq 50$

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