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
d) 1
Answer (1)
When vectors a and b are parallel, the angle between them is either 0 or 180 degrees.
The magnitude of the cross product ∣ a × b ∣ is 0, making a × b the zero vector.
The dot product a ⋅ b is equal to ∣ a ∣∣ b ∣ cos ( θ ) , which is not necessarily 0.
Therefore, a ⋅ b is equal to a × b , which is the zero vector. a × b
Explanation
Understanding Parallel Vectors When two vectors, a and b , are parallel, it means they point in the same direction or in opposite directions. The angle θ between them is either 0 degrees or 180 degrees. The dot product of two vectors is defined as a ⋅ b = ∣ a ∣∣ b ∣ cos ( θ ) , and the magnitude of the cross product is ∣ a × b ∣ = ∣ a ∣∣ b ∣ sin ( θ ) .
Calculating the Cross Product Since a and b are parallel, θ is either 0 or 180 degrees. Therefore, sin ( θ ) = sin ( 0 ) = 0 or sin ( θ ) = sin ( 18 0 ∘ ) = 0 . This means that the magnitude of the cross product is ∣ a × b ∣ = ∣ a ∣∣ b ∣ ( 0 ) = 0 . The cross product a × b is a vector with magnitude 0, which is the zero vector.
Calculating the Dot Product Now let's consider the dot product a ⋅ b = ∣ a ∣∣ b ∣ cos ( θ ) . If θ = 0 , then cos ( 0 ) = 1 , and a ⋅ b = ∣ a ∣∣ b ∣ . If θ = 18 0 ∘ , then cos ( 18 0 ∘ ) = − 1 , and a ⋅ b = − ∣ a ∣∣ b ∣ . In either case, the dot product is a scalar value that is not necessarily equal to 0 unless either a or b is the zero vector. However, the magnitude of the cross product, ∣ a × b ∣ , is 0. Therefore, a × b is the zero vector.
Conclusion Since a × b is the zero vector, we can say that a ⋅ b is equal to a × b , where a × b represents the zero vector.
Examples
Understanding parallel vectors is crucial in many real-world applications, such as navigation and physics. For example, when an airplane flies in a straight line at a constant speed, its velocity vector is parallel to its displacement vector. The dot product can then be used to calculate the work done by the airplane's engine, while the cross product helps determine if there are any rotational forces acting on the plane.
