If the two vectors are given as $\vec{a}$ and $\vec{b}$ then its dot product is expressed as a.b. Suppose these two vectors are separated by angle $\theta$. To know what's the angle measurement we solve with the below formula

The angle between two vectors formula is given by

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

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Given: $\vec{a}$ = 3i + 4j - k and $\vec{b}$ = 2i - j + k

The dot product is given by

a.b = (3i + 4j - k)(2i - j + k)

= (3)(2) + (4)(-1) + (-1)(1)

= 6-4-1

= 1

The magnitude of vectors is given by

|a| = $\sqrt{3^2 + 4^2 + (-1)^2}$ = $\sqrt{26}$ = 5.09

|b| = $\sqrt{2^2 + (-1)^2 + (1)^2}$ = $\sqrt{6}$ = 2.449

The angle between two vectors is

$\theta$ = cos

= cos

= cos

= cos

= 85.37

Given: $\vec{a}$ = 5i - j + k and $\vec{b}$ = i + j - k

The dot product is given by

a.b = (5i - j + k)(i + j - k)

= (5)(1) + (-1)(1) + (1)(-1)

= 5-1-1

= 3

The magnitude of vectors is given by

|a| = $\sqrt{5^2 + (-1)^2 + (1)^2}$ = $\sqrt{26}$ = 5.09

|b| = $\sqrt{1^2 + (1)^2 + (-1)^2}$ = $\sqrt{3}$ = 1.73

The angle between two vectors is

$\theta$ = cos

= cos

= cos

= cos

= 68.16