Find the angle between the vectors a = 2i + 3j + k and b = i - 2j + 3k.

Get the complete, step-by-step math solution for: "Find the angle between the vectors a = 2i + 3j + k and b = i - 2j + 3k....". Powered by SolveForX AI math tutor.

Step-by-Step Solution

Step 1: Recall the formula for the angle between two vectors

The angle $\theta$ between two vectors \mathbf{a} and \mathbf{b} can be found using the dot product formula. This formula relates the dot product of the vectors to the product of their magnitudes and the cosine of the angle between them.

Step 2: Calculate the dot product of the vectors

The dot product of two vectors \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} and \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} is given by a_1b_1 + a_2b_2 + a_3b_3$. We substitute the components of the given vectors.

Step 3: Calculate the magnitudes of the vectors

The magnitude of a vector \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k} is given by \sqrt{v_1^2 + v_2^2 + v_3^2}$. We calculate the magnitudes for both vectors \mathbf{a} and \mathbf{b}$.

Step 4: Substitute values into the formula and solve for the angle

Now we substitute the calculated dot product and magnitudes into the formula for \cos \theta$. After simplifying, we take the inverse cosine to find the angle \theta$.