The angle between the vectors (1, 0, −1, 3) and (1, √ 3, 3, −3) in R^4 is aπ. Find a.
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Example 1. Find the angle between two vectors a = {3; 4} and b = {4; 3}.
Solution: calculate dot product of vectors:
a·b = 3 · 4 + 4 · 3 = 12 + 12 = 24.
Calculate vectors magnitude:
|a| = √32 + 42 = √9 + 16 = √25 = 5
|b| = √42 + 32 = √16 + 9 = √25 = 5
Calculate the angle between vectors:
cos α = a · b = 24 = 24 = 0.96|a| · |b|5 · 525
Example 2. Find the angle between two vectors a = {7; 1} and b = {5; 5}.
Solution: calculate dot product of vectors:
a·b = 5 · 7 + 1 · 5 = 35 + 5 = 40.
Calculate vectors magnitude:
|a| = √72 + 12 = √49 + 1 = √50 = 5√2
|b| = √52 + 52 = √25 + 25 = √50 = 5√2
Calculate the angle between vectors:
cos α = a · b = 40 = 40 = 4 = 0.8|a| · |b|5√2 · 5√2505
Examples of spatial tasks
Example 3. Find the angle between two vectors a = {3; 4; 0} and b = {4; 4; 2}.
Solution: calculate dot product of vectors:
a·b = 3 · 4 + 4 · 4 + 0 · 2 = 12 + 16 + 0 = 28.
Calculate vectors magnitude:
|a| = √32 + 42 + 02 = √9 + 16 = √25 = 5
|b| = √42 + 42 + 22 = √16 + 16 + 4 = √36 = 6
Calculate the angle between vectors:
cos α = a · b = 28 = 14|a| · |b|5 · 615
Example 4. Find the angle between two vectors a = {1; 0; 3} and b = {5; 5; 0}.
Solution: calculate dot product of vectors:
a·b = 1 · 5 + 0 · 5 + 3 · 0 = 5.
Calculate vectors magnitude:
|a| = √12 + 02 + 32 = √1 + 9 = √10
|b| = √52 + 52 + 02 = √25 + 25 = √50 = 5√2
Calculate the angle between vectors:
cos α = a · b = 5 = 1 = √5 = 0.1√5|a| · |b|√10 · 5√22√510
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