Find the angle between two vectors A-2-3-4k- and B-5-2-4k-

Consider the vector #160; A⃗ = 2î + 3ĵ 4k̂ And B⃗ = 5î + 2ĵ + 4k̂ Now, as we know A⃗.B⃗=ABcosθ θ is the angle between A⃗ and B ...

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Byju's Answer

Standard XII

Mathematics

Applications of Dot Product

Find the angl...

Question

Find the angle between two vectors $\to A = 2^i + 3^j - 4^k$ and $\to B = 5^i + 2^j + 4^k$ .

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Solution

Consider the vector
$\to A = 2^i + 3^j - 4^k$
And $\to B = 5^i + 2^j + 4^k$
Now, as we know
$\to A . \to B = A B cos θ$
$θ$ is the angle between $\to A$ and $\to B$
Therefore,
$cos θ = \frac{\to A . \to B}{A B}$
$cos θ = \frac{(2^i + 3^j - 4^k) . (5^i + 2^j + 4^k)}{\sqrt{2^{2} + 3^{2} + {(- 4)}^{2}} \sqrt{5^{2} + 2^{2} + 4^{2}}}$
$= \frac{(10 + 6 - 16)}{\sqrt{2^{2} + 3^{2} + {(- 4)}^{2}} \sqrt{5^{2} + 2^{2} + 4^{2}}} = 0$
Therefore, $cos θ = 0$
$θ = 90^{\circ}$
Hence angle between the vectors is $90^{\circ}$ .

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Find the angle between two vectors →A=2^i+3^j−4^k and →B=5^i+2^j+4^k.

Find the angle between two vectors $\to A = 2^i + 3^j - 4^k$ and $\to B = 5^i + 2^j + 4^k$ .