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

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 $\theta$.

More from this chapter

• 3. If G is the centroid of Delta ABC, prove that GA + GB + GC = 0. Further if G_1 be the centroid of another Delta PQR, show that AP + BQ + CR = 3 GG_1.
• **Type I** 1. The position vector of four points A, B, C, D are a, b, 2 a + 3 b and a - 2 b respectively. Express the vectors AC, DB, BC and CA in terms of a an
• Let a = i + j + k, b = 2 i + 2 j + k and d = a × b. If c is a vector such that a · c = | c|, | c - 2 a|² = 8 and the angle between d and c is (π)/(4), then |10
• Consider two vectors u=3i-j and v=2i+j-, > 0. The angle between them is given by cos^(-1) ((√5)/(2√7)). Let v=v_1+v_2 where v_1 is parallel to u and v_2 is perp
• If the components of a=α i+β j+ k along and perpendicular to b= i+ j- k respectively, are (16)/(11)(3 i+ j- k) and (1)/(11)(-4 i-5 j-17 k), then α²+β²+ ² is equ
• Let a = i + 2 j + k and b = 2 i + j - k. Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a. Then such a vector c is:
• Let the point A divide the line segment joining the points P(-1,-1,2) and Q(5,5,10) internally in the ratio r:1 (r>0). If O is the origin and ( OQ · OA - (1)/(5
• Let a be a unit vector perpendicular to the vectors b= i-2 j+3 k and c=2 i+3 j- k, and makes an angle of cos^(-1)(-(1)/(3)) with the vector i+ j+ k. If a makes

All Vector Algebra solutions →