Consider vectors u = ⟨2, 1⟩ and v = ⟨4, –1⟩ with the angle between them equal to 40°. What are the scalar projections u on v and v on u?
Options:
1) u v = 1.33 and v u = 1.33
2) u v = 1.33 and v u = 3.16
3) u v = 1.71 and v u = 1.33
4) u v = 1.71 and v u = 3.16
Answer (2)
The scalar projection of vector u on vector v is approximately 1.71, while the scalar projection of vector v on vector u is approximately 3.16. Therefore, option 4 is the correct choice. This is calculated using the dot product and magnitudes of the vectors.
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To find the scalar projections of one vector onto another, we use the dot product and the cosine of the angle between them. The scalar projection of vector u onto vector v and vector v onto vector u can be computed using the following formulas:
Scalar Projection of u onto v (u_v) :
u v = ∣∣ v ∣∣ u ⋅ v
Here, u ⋅ v is the dot product of u and v , and ∣∣ v ∣∣ is the magnitude of vector v .
Calculating u ⋅ v :
u ⋅ v = 2 × 4 + 1 × ( − 1 ) = 8 − 1 = 7
Finding the magnitude of v ∣∣ v ∣∣ :
∣∣ v ∣∣ = 4 2 + ( − 1 ) 2 = 16 + 1 = 17
Substitute these values into the formula:
u v = 17 7 ≈ 1.71
Scalar Projection of v onto u (v_u) :
v u = ∣∣ u ∣∣ u ⋅ v
Finding the magnitude of u ∣∣ u ∣∣ :
∣∣ u ∣∣ = 2 2 + 1 2 = 4 + 1 = 5
Substitute into the formula:
v u = 5 7 ≈ 3.13
Based on these calculations, the scalar projections are:
Scalar Projection of u onto v : 1.71
Scalar Projection of v onto u : 3.13
Thus, the closest option is:
Option 4: u v = 1.71 and v u = 3.16 . It approximates our calculations closely.
