Vector Algebra
Angle between vectors
Grade None
Question:
<p>As <span>\(\vec{a} = 8\vec{b}\)</span>, we have <span>\(\vec{c} = -7\vec{b}\)</span>. Therefore, <span>\(\vec{a}\)</span> and <span>\(\vec{b}\)</span> are like vectors and <span>\(\vec{b}\)</span> and <span>\(\vec{c}\)</span> are unlike. This implies that <span>\(\vec{a}\)</span> and <span>\(\vec{c}\)</span> will be unlike. The angle between <span>\(\vec{a}\)</span> and <span>\(\vec{c}\)</span> is equal to:</p>
<p>\(0\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{\pi}{4}\)</p>
<p>\(\pi\)</p>
Step-by-Step Solution
Key Concept: When vectors are expressed as scalar multiples of a common vector, their relative directions are determined by the signs of the scalars. If both scalars are positive, vectors point the same way (like); if opposite signs, they point opposite ways (unlike). The angle between two vectors depends on whether they're like or unlike vectors.
Step 1: Analyze the given relationships. Given: a = 8 b and c = -7 b Step 2: Determine the direction of each vector. Since a = 8 b with positive scalar 8, vectors a and b point in the same direction (like vectors). Step 3: Determine relationship between b and c . Since c = -7 b with negative scalar -7, vectors c and b point in opposite directions (unlike vectors). Step 4: Find the angle between a and c . Since a points in the same direction as b , and c points opposite to b , then a and c point in opposite directions (unlike vectors). For two collinear unlike vectors, the angle between them is 180° (or π radians). ∴ The angle between a and c is 180° (or π)
Correct Answer: D