If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between them?
Practice Questions
Q1
If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between them?
0 degrees
90 degrees
45 degrees
60 degrees
Questions & Step-by-Step Solutions
If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between them?
Step 1: Identify the components of vector A and vector B. Vector A = 2i + 3j means A has components 2 (i) and 3 (j). Vector B = 4i + 5j means B has components 4 (i) and 5 (j).
Step 2: Calculate the dot product of vectors A and B. The dot product A · B = (2 * 4) + (3 * 5).
Step 3: Perform the multiplication: 2 * 4 = 8 and 3 * 5 = 15. Now add them together: 8 + 15 = 23. So, A · B = 23.
Step 4: Calculate the magnitude of vector A. The magnitude |A| = √(2^2 + 3^2). Calculate 2^2 = 4 and 3^2 = 9, then add them: 4 + 9 = 13. Now take the square root: |A| = √13.
Step 5: Calculate the magnitude of vector B. The magnitude |B| = √(4^2 + 5^2). Calculate 4^2 = 16 and 5^2 = 25, then add them: 16 + 25 = 41. Now take the square root: |B| = √41.
Step 6: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A| * |B|). Substitute the values: cos(θ) = 23 / (√13 * √41).
Step 7: Calculate the denominator: √13 * √41 = √(13 * 41). Calculate 13 * 41 = 533, so the denominator is √533.
Step 8: Now calculate cos(θ): cos(θ) = 23 / √533.
Step 9: Use a calculator to find the value of cos(θ) and then find θ by taking the inverse cosine (arccos) of that value.
Step 10: The result will give you the angle θ in degrees. In this case, it is approximately 60 degrees.
