Vectors
Vectors
Allen Star Batch
Grade 12
Question:
If vectors $\vec{b} = (\tan\alpha, -1, 2\sqrt{\sin\frac{\alpha}{2}})$ and $\vec{c} = (\tan\alpha, \tan\alpha, -\frac{3}{\sqrt{\sin\alpha/2}})$ are orthogonal and vector $\vec{a} = (1, 3, \sin2\alpha)$ makes an obtuse angle with the z-axis then:
$\alpha = \tan^{-1}(-2)$
$\alpha = \tan^{-1}(-3)$
$\alpha = \tan^{-1}(2)$
None of these
Step-by-Step Solution
Key Concept: Orthogonality and dot product signs constrain the geometric configuration of vectors.
Given $\vec{b} \cdot \vec{c} = 0$ and $\vec{a} \cdot \vec{c} < 0$. The geometry shows these three vectors with specific orthogonality and sign constraints that determine their relative configuration in space, illustrated by the diagram showing point $R(\vec{r})$ relative to positions $A$ and $B$.
Correct Answer: 1