Let's analyze the given problem step by step.
When we talk about the dot product of two vectors, it can be described mathematically as:
[tex]\[ \vec{v_1} \cdot \vec{v_2} = |\vec{v_1}| |\vec{v_2}| \cos \theta \][/tex]
Here:
- [tex]\( |\vec{v_1}| \)[/tex] is the magnitude of the first vector
- [tex]\( |\vec{v_2}| \)[/tex] is the magnitude of the second vector
- [tex]\( \theta \)[/tex] is the angle between the two vectors
The given condition is that the dot product of two non-zero vectors [tex]\( \vec{v_1} \)[/tex] and [tex]\( \vec{v_2} \)[/tex] is zero:
[tex]\[ \vec{v_1} \cdot \vec{v_2} = 0 \][/tex]
Given that [tex]\( |\vec{v_1}| \)[/tex] and [tex]\( |\vec{v_2}| \)[/tex] are both non-zero, the only way for the dot product to equal zero is for [tex]\( \cos \theta \)[/tex] to equal zero. This occurs when:
[tex]\[ \theta = 90^\circ \text{ or } \theta = \pi/2 \text{ radians} \][/tex]
In other words, [tex]\( \theta = 90^\circ \)[/tex] means that the vectors [tex]\( \vec{v_1} \)[/tex] and [tex]\( \vec{v_2} \)[/tex] are perpendicular to each other. Therefore, the correct interpretation of the given condition is:
D. [tex]\( \vec{v_1} \)[/tex] is perpendicular to [tex]\( \vec{v_2} \)[/tex].
