Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|co
Practice Questions
Q1
Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|cos(θ).
60°
45°
30°
90°
Questions & Step-by-Step Solutions
Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|cos(θ).
Step 1: Identify the vectors A and B. A = 4i + 3j and B = 1i + 2j.
Step 2: Calculate the dot product A · B. Use the formula A · B = Ax * Bx + Ay * By. Here, Ax = 4, Ay = 3, Bx = 1, By = 2. So, A · B = 4*1 + 3*2 = 4 + 6 = 10.
Step 3: Calculate the magnitude of vector A, |A|. Use the formula |A| = √(Ax^2 + Ay^2). Here, |A| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.
Step 4: Calculate the magnitude of vector B, |B|. Use the formula |B| = √(Bx^2 + By^2). Here, |B| = √(1^2 + 2^2) = √(1 + 4) = √5.
Step 5: Use the formula A · B = |A||B|cos(θ) to find cos(θ). We have A · B = 10, |A| = 5, and |B| = √5. So, 10 = 5 * √5 * cos(θ).
Step 6: Rearrange the equation to find cos(θ). So, cos(θ) = 10 / (5 * √5) = 2 / √5.
Step 7: Use the inverse cosine function to find θ. θ = cos⁻¹(2/√5). This gives θ ≈ 45°.
Dot Product – Understanding the dot product of two vectors and its relation to the angle between them.
Magnitude of Vectors – Calculating the magnitude of vectors using the Pythagorean theorem.
Cosine of Angle – Using the cosine function to relate the dot product and magnitudes of vectors to find the angle.
