If [tex]\vec{a}[/tex] is parallel to [tex]\vec{b}[/tex], then what is the value of [tex]\vec{a} \cdot \vec{b}[/tex] equal to?
a) [tex]\vec{a} \times \vec{b}[/tex]
b) [tex]90^{\circ}[/tex]
c) 0
d) 1
Answer (2)
The dot product of two parallel vectors a and b is defined as ∣ a ∣∣ b ∣ cos ( θ ) , where θ is the angle between the vectors.
When the vectors are parallel, the angle θ is either 0° or 180°.
Therefore, cos ( 0 ∘ ) = 1 and cos ( 18 0 ∘ ) = − 1 , which means the dot product is either ∣ a ∣∣ b ∣ or − ∣ a ∣∣ b ∣ .
None of the given options match this result, indicating that none of them are correct.
Explanation
Understanding Parallel Vectors and Dot Product When two vectors, a and b , are parallel, the angle θ between them is either 0 degrees or 180 degrees. The dot product of two vectors is defined as:
Dot Product Formula a ⋅ b = ∣ a ∣∣ b ∣ cos ( θ )
where ∣ a ∣ and ∣ b ∣ are the magnitudes of the vectors a and b , respectively, and θ is the angle between them.
Applying Parallel Condition If a and b are parallel, then θ is either 0 ∘ or 18 0 ∘ .
If θ = 0 ∘ , then cos ( 0 ∘ ) = 1 , so a ⋅ b = ∣ a ∣∣ b ∣ .
If θ = 18 0 ∘ , then cos ( 18 0 ∘ ) = − 1 , so a ⋅ b = − ∣ a ∣∣ b ∣ .
Analyzing the Result The dot product a ⋅ b can be either ∣ a ∣∣ b ∣ or − ∣ a ∣∣ b ∣ , depending on whether the vectors are in the same or opposite directions. This means the dot product is equal to the product of their magnitudes (if in the same direction) or the negative of the product of their magnitudes (if in opposite directions).
Evaluating the Options Now, let's examine the given options: a) a × b : This is the cross product of a and b , which results in a vector, not a scalar like the dot product. So, this is not the correct answer. b) 9 0 ∘ : This is an angle, not a value of a dot product. c) 0: The dot product is 0 when the vectors are orthogonal (perpendicular), not parallel. d) 1: The dot product is 1 only in specific cases where the magnitudes of both vectors and the cosine of the angle between them multiply to 1. This is not generally true for all parallel vectors.
Conclusion Since the dot product a ⋅ b is equal to ∣ a ∣∣ b ∣ or − ∣ a ∣∣ b ∣ , none of the provided options ( a × b , 9 0 ∘ , 0, or 1) are generally equal to a ⋅ b when a and b are parallel. Therefore, none of the given options are correct.
Examples
In physics, when calculating the work done by a force, if the force and displacement are parallel, the work done is simply the product of the magnitudes of the force and displacement. If they are anti-parallel, the work done is the negative of the product of their magnitudes. This concept is crucial in understanding energy transfer in various physical systems.
When two vectors a and b are parallel, their dot product a ⋅ b is equal to either ∣ a ∣∣ b ∣ or − ∣ a ∣∣ b ∣ , but none of the provided options are correct. Thus, the options provided do not match the expected answers based on the properties of dot products and parallel vectors.
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