Given u = (3,2), v = (1, 5), and the angle between the vectors is 45ยฐ, what is u ยท v?
Answer (2)
The dot product of vectors u = ( 3 , 2 ) and v = ( 1 , 5 ) , calculated using both components and magnitudes, is 13 .
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Calculate the dot product using the components: u โ
v = ( 3 ) ( 1 ) + ( 2 ) ( 5 ) = 13 .
Calculate the magnitudes of vectors u and v : โฃ u โฃ = 13 โ and โฃ v โฃ = 26 โ .
Calculate the dot product using magnitudes and the angle: u โ
v = โฃ u โฃโฃ v โฃ cos ( 4 5 โ ) = 13 .
The dot product of u and v is 13 โ .
Explanation
Problem Analysis We are given two vectors, u = ( 3 , 2 ) and v = ( 1 , 5 ) , and the angle between them is 4 5 โ . We need to find the dot product u โ
v . We can calculate the dot product in two ways: using the magnitudes of the vectors and the angle between them, or using the components of the vectors.
Dot Product Using Components First, let's calculate the dot product using the components of the vectors: u โ
v = ( 3 ) ( 1 ) + ( 2 ) ( 5 ) = 3 + 10 = 13
Calculate Magnitudes Now, let's calculate the magnitudes of the vectors u and v :
โฃ u โฃ = 3 2 + 2 2 โ = 9 + 4 โ = 13 โ โฃ v โฃ = 1 2 + 5 2 โ = 1 + 25 โ = 26 โ
Dot Product Using Magnitudes and Angle The dot product can also be calculated using the formula: u โ
v = โฃ u โฃโฃ v โฃ cos ( ฮธ ) where ฮธ is the angle between the vectors. In this case, ฮธ = 4 5 โ , so cos ( 4 5 โ ) = 2 2 โ โ .
u โ
v = 13 โ โ
26 โ โ
cos ( 4 5 โ ) = 13 โ โ
26 โ โ
2 2 โ โ = 13 โ โ
13 โ
2 โ โ
2 2 โ โ = 13 โ โ
13 โ โ
2 โ โ
2 2 โ โ = 13 โ
2 2 โ = 13
Final Answer Both methods give us the same result, so the dot product of u and v is 13.
Examples
The dot product is incredibly useful in physics and engineering. For example, if you're pushing a lawnmower, the dot product can tell you how much of your force is actually going into moving the mower forward versus how much is wasted. If the force vector and the displacement vector are F and d respectively, then the work done is W = F โ
d = โฃ F โฃโฃ d โฃ cos ( ฮธ ) , where ฮธ is the angle between the force and displacement. This helps optimize the efficiency of energy transfer in various mechanical systems.
