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Applications of Dot Product

If A⃗=2i⃗+k...

Question

If $\to A = 2 \to i + \to k, \to B = \to i + \to j + \to k$ and $\to C = 4 \to i - 3 \to j + 7 \to k$ . Determine a vector $\to R$ satisfying $\to R \times \to B = \to C \times \to B$ and $\to R \cdot \to A = 0$ .

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Solution

$\begin{matrix} \to R \times \to B = \to C \times \to B = 0 (\to R - \to C) \times \to B = 0 \to R - \to C = λ \to B \to R = λ \to B + \to C \to R . \to A = λ \to B . \to A + \to C . \to A 0 = λ (3) + 15 ∴ λ = - 5 \to R = - 5 \to B + \to C \to R = - 5 (^i +^j +^k) + 4^i - 3^j + 7^k \to R = -^i - 8^j + 2^k \end{matrix}$

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Similar questions

\to A = 2 \to i + \to k, \to B = \to i + \to j + \to k,

\to C = 4 \to i - 3 \to j + 7 \to k

Determine a vector

\to R

\to R \times \to B = \to C \times \to B

\to R . \to A = 0

\to r = 3 \to i + 2 \to j - 5 \to k

\to a = 2 \to i - \to j + \to k

\to b = \to i + 3 \to j - 2 \to k

\to c = - 2 \to i + \to j - 3 \to k

\to r = λ \to a + μ \to b + ν \to c

\to a = \to i + \to j - \to k, \to b = 5 \to i - 3 \to j - 3 \to k, \to c = 3 \to i - \to j + 2 \to k

\to r

is colinear with

\to c

| \to r | = ∣ ∣ \to a + \to b ∣ ∣

\to r

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

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

\to a = \to i + \to j + \to k, \to c = \to j - \to k

\to b

is a vector satisfying

\to a \times \to b = \to c

\to a . \to b = 3

\to b

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