Given $u =\langle 1,3\rangle, v =\langle 2,1\rangle$, and $\cos (\theta)=\frac{\sqrt{2}}{2}$, where $\theta$ is the angle between the vectors, what is the scalar projection $u _{ v }$ and the dot product $u \cdot v$ ?
A. $u _{ v }=1.41$ and $u \cdot v =4.47$
B. $u _{ v }=2.24$ and $u \cdot v =5.00$
C. $u _{ v }=2.24$ and $u \cdot v =7.07$
D. $u _{ v }=7.07$ and $u \cdot v =15.81$
Answer (2)
The scalar projection of vector u onto vector v is approximately 2.24 , and the dot product of the vectors u and v is 5.00 . Therefore, the correct answer is option B. The calculations involved finding the dot product and the magnitudes of the vectors.
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Calculate the dot product of vectors u and v : u ⋅ v = ( 1 ) ( 2 ) + ( 3 ) ( 1 ) = 5 .
Calculate the magnitude of vector v : ∣∣ v ∣∣ = 2 2 + 1 2 = 5 ≈ 2.236 .
Calculate the scalar projection of u onto v : u v = ∣∣ v ∣∣ u ⋅ v = 5 5 = 5 ≈ 2.236 .
The scalar projection is approximately 2.24 and the dot product is 5.00 , so the answer is: u v = 2.24 and u ⋅ v = 5.00 .
Explanation
Problem Analysis We are given two vectors u = ⟨ 1 , 3 ⟩ and v = ⟨ 2 , 1 ⟩ , and the cosine of the angle between them, cos ( θ ) = 2 2 . We need to find the scalar projection of u onto v , denoted as u v , and the dot product of u and v , denoted as u ⋅ v .
Calculate the dot product First, let's calculate the dot product u ⋅ v using the components of the vectors: u ⋅ v = ( 1 ) ( 2 ) + ( 3 ) ( 1 ) = 2 + 3 = 5
Calculate the magnitude of v Next, let's calculate the magnitude of vector v :
∣∣ v ∣∣ = ( 2 ) 2 + ( 1 ) 2 = 4 + 1 = 5 ≈ 2.236
Calculate the scalar projection Now, we can calculate the scalar projection of u onto v using the formula: u v = ∣∣ v ∣∣ u ⋅ v = 5 5 = 5 5 5 = 5 ≈ 2.236
Alternative dot product calculation Alternatively, we can calculate the dot product using the magnitudes of the vectors and the cosine of the angle between them: u ⋅ v = ∣∣ u ∣∣∣∣ v ∣∣ cos ( θ ) where ∣∣ u ∣∣ = 1 2 + 3 2 = 10 and ∣∣ v ∣∣ = 2 2 + 1 2 = 5 and cos ( θ ) = 2 2 .
u ⋅ v = 10 ⋅ 5 ⋅ 2 2 = 50 ⋅ 2 2 = 5 2 ⋅ 2 2 = 2 5 ⋅ 2 = 5
Final Answer Therefore, the scalar projection u v ≈ 2.24 and the dot product u ⋅ v = 5.00 .
Examples
Understanding scalar projections and dot products is crucial in physics, especially when analyzing forces acting at angles. For instance, if you're pulling a sled with a rope at an angle, the scalar projection tells you how much of your force is actually moving the sled forward. The dot product helps calculate the work done by the force. These concepts are also used in computer graphics to determine lighting and shading effects on 3D objects, making scenes appear more realistic.
