Question

Find the angle between the vectors $\hat i-2\hat j+3 \hat k\ and 3\hat i-2\hat j+ \hat k$.

Answer 1

Let the two vectors a and b.

Now, Whenever any vector is given in terms of i, j and k cap, then remember, you should apply the formula for dot product since, it is little easy for the calculation.

But you can apply any either cross or dot as you like but dot is little easy.

Applying dot product.

|a.b| = |a||b|Cosθ

Now, |a|² = (1)² + (-2)² + (3)²

∴ |a|² = 1 + 4 + 9

∴ |a|² = 14

∴ |a| = ± √14

Magnitude cannot have negative value, therefore, |a| = + √14

Similarly,

|b|² = (3)² + (-2)² + (1)²

∴ |b|² = 14

∴ |b| = ± √14

Magnitude cannot have negative value, therefore, |b| = + √14

Now, |a.b| = (1 × 3) + (-2 × -2) + (3 × 1)

∴  |a.b| = 3 + 4 + 3

∴ |a.b| = 10

Now, Putting all these values in the formula of dot product.

We get,

10 = √14 × √14 Cosθ

Cosθ = 5/7

Therefore, θ = Cos⁻¹(5/7)

Hence, angle between the two vectors is Cos⁻¹(5/7).

Hope it helps.