Vector Algebra
Angle Between Vectors
Grade None
Question:
<p>The non-zero vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are related by \(\vec{a} = 8\vec{b}\) and \(\vec{c} = -7\vec{b}\). Then the angle between \(\vec{a}\) and \(\vec{c}\) is</p>
<p>0</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\pi\)</p>
Step-by-Step Solution
Key Concept: When two vectors are scalar multiples of the same vector, the angle between them depends on the signs of the scalars. Positive scalar multiples point in the same direction (0°), while opposite sign scalars point in opposite directions (180°).
Step 1: Express both vectors in terms of $\vec{b}$ Given: $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$ Step 2: Analyze the relationship between $\vec{a}$ and $\vec{c}$ We can write: $\vec{c} = -7\vec{b} = -\frac{7}{8}(8\vec{b}) = -\frac{7}{8}\vec{a}$ Since $\vec{c} = k\vec{a}$ where $k = -\frac{7}{8} < 0$, the vectors are antiparallel (point in opposite directions). Step 3: Determine the angle When one vector is a negative scalar multiple of another, they point in exactly opposite directions. ∴ The angle between $\vec{a}$ and $\vec{c}$ is 180° or π radians
Correct Answer: D