If a - a 1- a 2- a 3k- - b - b 1- b 2- b 3k- - c - c 1- c 2- c 3k- and -3 a - b - 3 b - c - 3 c - a -loverrightarrow clcdot is i- loverrightarrow clcdot hat j- loverrightarrow clcdot k lend vmatrix then the value of -4 is

A=a_1î+a_2ĵ+a_3k̂;b=b_1î+b_2ĵ+b_3k̂;c=c_1î+c_2ĵ+c_3k̂ L.H.S. =[3a+b 3b+c 3c+a] =[3a 3b 3c]+[bca] =3^3[abc]+abc] =28[abc] nbsp; ...

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Applications Scalar Triple Product

If a = a 1î+ ...

Question

If $\to a = a_{1}^i + a_{2}^j + a_{3}^k; \to b = b_{1}^i + b_{2}^j + b_{3}^k;$ $\to c = c_{1}^i + c_{2}^j + c_{3}^k$ and $[3 \to a + \to b, 3 \to b + \to c, 3 \to c + \to a]$
$= λ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a \cdot^i & \to a \cdot^j & \to a \cdot^k \to b \cdot^i & \to b \cdot^j & \to b \cdot^k \to c \cdot^i & \to c \cdot^j & \to c \cdot^k \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
then the value of $\frac{λ}{4}$ is

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\to a = a_{1}^i + a_{2}^j + a_{3}^k; \to b = b_{1}^i + b_{2}^j + b_{3}^k;

\to c = c_{1}^i + c_{2}^j + c_{3}^k

[3 \to a + \to b, 3 \to b + \to c, 3 \to c + \to a]

= λ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \to a \cdot^i & \to a \cdot^j & \to a \cdot^k \to b \cdot^i & \to b \cdot^j & \to b \cdot^k \to c \cdot^i & \to c \cdot^j & \to c \cdot^k \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

\frac{λ}{4}

\to a = a_{1}^i + a_{2}^j + a_{3}^k, \to b = b_{1}^i + b_{2}^j + b_{3}^k, \to c = c_{1}^i + c_{2}^j + c_{3}^k

∣ ∣ \to c ∣ ∣ = 1

(\to a \times \to b) \times \to c = 0

{∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2} =

\to a, \to b, \to c

a_{1}^i + a_{2}^j + a_{3}^k, b_{1}^i + b_{2}^j + b_{3}^k, c_{1}^i + c_{2}^j + c_{3}^k

\to a \times (\to b + \to c) = \to a \times \to b + \to a \times \to c

\to b + \to c

\to a

\to a = 2^i - 3^j + 4^k \to b = 3^i - 3^j + 3^k \to c =^i +^j + 3^k

\to a = a_{1}^i + a_{2}^j - a_{3}^k

\to b = b_{1}^i + b_{2}^j - b_{3}^k

\to a \times \to b

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