Question

The angle between the two vectors a=3i+2j+4k and b=2i+j-2k is equal to

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

Answer:

The angle between the given two vectors is 90°.

Step-by-step explanation:

Given two vectors $\vec a=3\hat i+2\hat j+4\hat k$ and $\vec b=2\hat i+\hat j-2\hat k$

The dot product of two vectors is given by

$\vec a.\vec b=\big|\vec a\big|\big|\vec b\big|~cos\theta\implies cos\theta=\frac{\vec a.\vec b}{\big|\vec a\big|\big|\vec b\big|}$                             ...(1)

Magnitude of vector a is

$\implies\big|\vec a\big|=\sqrt{3^{2} +2^{2} +4^{2}$

            $=\sqrt{9 +4 +16 }=\sqrt{29 }$                                     ...(2)

Magnitude of vector b is

$\implies\big|\vec b\big|=\sqrt{2^{2} +1^{2} +(-2)^{2}$

            $=\sqrt{4 +1 +4 }=\sqrt{9 }=3$                                  ...(3)

Product of vector a and vector b is

 $\vec a.\vec b=\big(3\hat i+2\hat j+4\hat k\big).\big(2\hat i+\hat j-2\hat k\big)$

$=3\times 2+2\times 1+4\times(-2)=6+2-8=0$                 ...(4)

Substituting (2), (3) and (4) in (1),

$cos\theta=\frac{0}{\sqrt{29}\times 3}=0\implies \theta=90^{o}$  

Therefore, the angle between the two vectors is 90°. They are perpendicular to each other.