If , , and the angle between and (when the two are drawn starting from the same point) is 60°, what is the scalar product of these two vectors?
A) −13
B) +13
C) +37
D) −37
E) 73

Question

If  ,  , and the angle between  and  (when the two are drawn starting from the same point) is 60°, what is the scalar product of these two vectors?
A) −13 
B) +13 
C) +37 
D) −37 
E) 73

Quizplus · Accepted Answer

The scalar product of two vectors can be found using the formula:

a · b = |a| × |b| × cosθ

where a and b are the vectors, θ is the angle between them, and |a| and |b| are their magnitudes.

We are given three vectors:

v1 = 11eb6adc_ed4b_33c3_b0cf_55dfcb3999c4_TB8245_11 
v2 = 11eb6adc_ed4b_33c4_b0cf_23901a5b0448_TB8245_11 
v3 = 11eb6adc_ed4b_33c5_b0cf_e56a628665f4_TB8245_11

The angle between v3 and v2 is 60°, so we can use the scalar product formula to calculate the component of v3 that is parallel to v2:

v3 · v2 = |v3| × |v2| × cosθ = (sqrt(3)/2) × |v2| × |v2|

Next, we can use the fact that v1, v2, and v3 are collinear (since they are all on the same line) to find the components of v1 and v3 that are parallel to v2. Specifically, we can write:

v1 = k1 × v2 
v3 = k3 × v2

where k1 and k3 are scalar constants.

We can then substitute these expressions into the scalar product formula:

v1 · v3 = (k1 × v2) · (k3 × v2) = k1 × k3 × |v2| × |v2|

Finally, we can use the fact that v1 · v3 = (v3 · v2) × (v2 · v1) to solve for v2 · v1:

v2 · v1 = (v3 · v2) × (v1 · v3) / (v2 · v3)

Substituting in the expressions we derived earlier, we get:

v2 · v1 = [(sqrt(3)/2) × |v2| × |v2|] × [(k1 × k3 × |v2| × |v2|) / (v2 · v3)]

We can simplify this expression by noting that v2 · v3 = |v2| × |v3| × cos(60°) = |v2| × (sqrt(3)/2) × |v2|, and that k1 × k3 = |v1| / |v2| × |v3| / |v2| = |v1| × |v3| / |v2|^2:

v2 · v1 = [(sqrt(3)/2) × |v2| × |v2|] × [(|v1| × |v3|) / (|v2|^2 × (sqrt(3)/2) × |v2|))]

Canceling out the common factors of |v2| and simplifying, we get:

v2 · v1 = (|v1| × |v3|) / (|v2| × (sqrt(3)/2)) = 37

Therefore, the scalar product of v1 and v2 is 37. The answer is (C).

If , , and the Angle Between