Suppose vector $u =\langle 1, \sqrt{3}\rangle, | v |=6$, and the angle between the vectors is $120^{\circ}$. What is $u \cdot v$?
Answer (2)
The dot product of the vectors u and v is calculated to be -6 by using the dot product formula, taking into account the magnitude of u , the magnitude of v , and the angle between them. The cosine of the angle 12 0 ∘ is -1/2, which leads to this result. Therefore, the final answer is -6.
;
Calculate the magnitude of vector u : ∣ u ∣ = 1 2 + ( 3 ) 2 = 2 .
Apply the dot product formula: u ⋅ v = ∣ u ∣∣ v ∣ cos ( θ ) .
Substitute the values: u ⋅ v = 2 ⋅ 6 ⋅ cos ( 12 0 ∘ ) = 12 ⋅ ( − 2 1 ) .
Calculate the dot product: u ⋅ v = − 6 . The final answer is − 6 .
Explanation
Problem Analysis We are given vector u = ⟨ 1 , 3 ⟩ , the magnitude of vector v as ∣ v ∣ = 6 , and the angle between u and v is 12 0 ∘ . We want to find the dot product u ⋅ v .
Dot Product Formula The dot product of two vectors can be calculated using the formula: u ⋅ v = ∣ u ∣∣ v ∣ cos ( θ ) where ∣ u ∣ and ∣ v ∣ are the magnitudes of the vectors u and v , respectively, and θ is the angle between them.
Magnitude of u First, we need to find the magnitude of vector u :
∣ u ∣ = 1 2 + ( 3 ) 2 = 1 + 3 = 4 = 2
Substitute Values Now, we can substitute the given values into the dot product formula: u ⋅ v = ∣ u ∣∣ v ∣ cos ( 12 0 ∘ ) = 2 ⋅ 6 ⋅ cos ( 12 0 ∘ ) We know that cos ( 12 0 ∘ ) = − 2 1 .
Calculate Dot Product Now, we calculate the dot product: u ⋅ v = 2 ⋅ 6 ⋅ ( − 2 1 ) = 12 ⋅ ( − 2 1 ) = − 6
Final Answer Therefore, the dot product of vectors u and v is -6.
Examples
The dot product is incredibly useful in physics and engineering. For example, if you're calculating the work done by a force, you use the dot product of the force vector and the displacement vector. Imagine pushing a box across a floor; the work done is maximized when you push in the direction of motion and minimized when you push perpendicular to it. The dot product captures this relationship perfectly, allowing engineers to design efficient systems and predict energy expenditure in various scenarios.
