Vectors a, b, c are inclined to each other at | vedclass

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(D) Given that $|a| = |b| = |c| = 2$ and the angle between any two vectors is $60^\circ$. Thus,$a \cdot b = b \cdot c = c \cdot a = |a||b| \cos 60^\ci

Vedclass · Accepted Answer

$-120$

Vedclass · Answer

(D) Given that $|a| = |b| = |c| = 2$ and the angle between any two vectors is $60^\circ$. Thus,$a \cdot b = b \cdot c = c \cdot a = |a||b| \cos 60^\circ = 2 	imes 2 	imes \frac{1}{2} = 2$. Now,consider the dot product: $(2a + 3b - 5c) \cdot (4a - 6b + 10c) = 8(a \cdot a) - 12(a \cdot b) + 20(a \cdot c) + 12(b \cdot a) - 18(b \cdot b) + 30(b \cdot c) - 20(c \cdot a) + 30(c \cdot b) - 50(c \cdot c)$. Since $a \cdot b = b \cdot a$,$a \cdot c = c \cdot a$,and $b \cdot c = c \cdot b$,the expression simplifies to: $8|a|^2 - 18|b|^2 - 50|c|^2 + (-12 + 12)(a \cdot b) + (20 - 20)(a \cdot c) + (30 + 30)(b \cdot c)$. $= 8(2)^2 - 18(2)^2 - 50(2)^2 + 60(b \cdot c)$. $= 8(4) - 18(4) - 50(4) + 60(2)$. $= 32 - 72 - 200 + 120$. $= -120$.

Vectors $a, b, c$ are inclined to each other at an angle of $60^\circ$. If $|a| = 2, |b| = 2$,and $|c| = 2$,then calculate the value of $(2a + 3b - 5c) \cdot (4a - 6b + 10c)$.

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