Solved Dot Product Problems
Learn dot product calculation through these solved examples. Each problem includes a step-by-step solution to help you understand the dot product definition and its mathematical properties.
Example 1: Basic 2D Dot Product
Calculate the dot product of vectors A = (3, 4) and B = (2, 5).
Solution:
A·B = AxBx + AyBy
A·B = (3 × 2) + (4 × 5)
A·B = 6 + 20 = 26
Example 2: Orthogonal Vectors
Show that vectors A = (2, 5) and B = (-5, 2) are orthogonal.
Solution:
Vectors are orthogonal if their dot product is zero.
A·B = (2 × -5) + (5 × 2)
A·B = -10 + 10 = 0
Since the dot product is zero, the vectors are orthogonal.
Example 3: Commutative Property
Verify the commutative property with vectors A = (1, 3, -2) and B = (4, -1, 5).
Solution:
Commutative property: A·B = B·A
Calculate A·B:
A·B = (1×4) + (3×-1) + (-2×5) = 4 - 3 - 10 = -9
Calculate B·A:
B·A = (4×1) + (-1×3) + (5×-2) = 4 - 3 - 10 = -9
Both calculations give the same result, verifying the commutative property.
Example 4: Angle Between Vectors
Find the angle between vectors A = (2, 3) and B = (4, 1).
Solution:
Using the geometric formula:
A·B = ||A|| ||B|| cosθ
First, calculate dot product:
A·B = (2×4) + (3×1) = 8 + 3 = 11
Calculate magnitudes:
||A|| = √(2² + 3²) = √(4+9) = √13 ≈ 3.606
||B|| = √(4² + 1²) = √(16+1) = √17 ≈ 4.123
Solve for cosθ:
cosθ = (A·B) / (||A|| ||B||) = 11 / (3.606 × 4.123) ≈ 11 / 14.87 ≈ 0.740
θ = arccos(0.740) ≈ 42.2°
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