If [tex]$\vec{a}$[/tex] is parallel to [tex]$\vec{b}$[/tex], then the value of [tex]$\vec{a} \cdot \vec{b}$[/tex] is equal to
a) [tex]$\vec{a} \times \vec{b}$[/tex]
b) [tex]$90^{\circ}$[/tex]
c) 0
Answer (2)
The dot product of two parallel vectors a and b is given by a ⋅ b = ∣ a ∣∣ b ∣ cos ( θ ) , where θ is the angle between the vectors.
When a and b are parallel, θ is either 0 ∘ or 18 0 ∘ .
If either a or b is the zero vector, then a ⋅ b = 0 .
Therefore, the value of a ⋅ b can be 0. 0
Explanation
Analyze the problem and options. When two vectors, a and b , are parallel, the angle θ between them is either 0 ∘ or 18 0 ∘ . The dot product of two vectors is defined as a ⋅ b = ∣ a ∣∣ b ∣ cos ( θ ) . Let's analyze each option:
a) a × b : The cross product of two parallel vectors is the zero vector, 0 . The dot product is a scalar, and it's not necessarily equal to the zero vector unless one or both of the vectors a or b are zero vectors. So, this option is not always true.
b) 9 0 ∘ : The dot product is a scalar value, not an angle. So, this option is incorrect.
c) 0: The dot product is zero only when the vectors are perpendicular (i.e., the angle between them is 9 0 ∘ ). Since a and b are parallel, their dot product is not necessarily zero unless one or both of the vectors are zero vectors and they are perpendicular. If the vectors are non-zero, the dot product will be non-zero.
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 ∣ .
Therefore, the dot product a ⋅ b is equal to ∣ a ∣∣ b ∣ or − ∣ a ∣∣ b ∣ , depending on whether the vectors point in the same or opposite directions.
Reconsider the options. The correct answer is that the value of a ⋅ b is not necessarily equal to a × b , 9 0 ∘ , or 0. However, the question seems to imply that one of the options is correct. Let's reconsider the options.
If the question is asking which of the options CAN be the value of a ⋅ b , then: a) a × b is a vector, while a ⋅ b is a scalar, so they cannot be equal. b) 9 0 ∘ is an angle, while a ⋅ b is a scalar, so they cannot be equal. c) 0 is a scalar, and a ⋅ b can be 0 if either a or b is the zero vector.
If a and b are parallel, then a = k b for some scalar k . Then a ⋅ b = ( k b ) ⋅ b = k ( b ⋅ b ) = k ∣ b ∣ 2 . If k = 0 or b = 0 , then a ⋅ b = 0 .
Final Analysis. Given that a is parallel to b , the angle between them is either 0 ∘ or 18 0 ∘ . The dot product a ⋅ b = ∣ a ∣∣ b ∣ cos ( θ ) .
If θ = 0 ∘ , a ⋅ b = ∣ a ∣∣ b ∣ .
If θ = 18 0 ∘ , a ⋅ b = − ∣ a ∣∣ b ∣ .
Option a) a × b is a vector, while a ⋅ b is a scalar. Also, since a and b are parallel, a × b = 0 . So, a ⋅ b can be 0, but it is not necessarily equal to a × b .
Option b) 9 0 ∘ is an angle, while a ⋅ b is a scalar. So, this is incorrect. Option c) 0 is a scalar. If either a or b is the zero vector, then a ⋅ b = 0 . So, this is possible.
If the vectors are non-zero, then the dot product is non-zero. However, if either vector is the zero vector, the dot product is zero. So, the value of a ⋅ b can be 0.
Final Answer. The value of a ⋅ b can be 0 if either a or b is the zero vector. Therefore, the answer is:
0
Examples
Dot products are used in physics to calculate work done by a force. If the force and displacement are parallel, the work done is the product of their magnitudes. If they are perpendicular, no work is done. This concept is crucial in understanding energy transfer in various physical systems.
The value of a ⋅ b is equal to 0 if either vector is the zero vector. For parallel vectors, the dot product depends on the angle between them, yielding values either positive or negative depending on their directions. Thus, the chosen option is 0 .
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