If a⃗ = b⃗ + c⃗, then is it true that |a⃗| = |b⃗| + |c⃗|? Justify your answer.
No, it is not always true that |a⃗| = |b⃗| + |c⃗| when a⃗ = b⃗ + c⃗.
By the triangle inequality, we have |b⃗ + c⃗| |b⃗| + |c⃗|. Since a⃗ = b⃗ + c⃗, this means |a⃗| |b⃗| + |c⃗|.
The equality |a⃗| = |b⃗| + |c⃗| holds only when the vectors b⃗ and c⃗ are collinear and point in the same direction. In general, when vectors are added, the magnitude of the resultant vector is less than or equal to the sum of individual magnitudes, depending on the angle between them.
Explanation
This question tests understanding of the triangle inequality for vectors. The context clearly states that |a⃗ + b⃗| |a⃗| + |b⃗|, with equality holding only when points are collinear. Students should recognize that vector magnitudes don't simply add like scalars unless the vectors are collinear and in the same direction.
Solution Steps
Step 1: Given a⃗ = b⃗ + c⃗, we need to find |a⃗| = |b⃗ + c⃗|
Step 2: Apply triangle inequality: |b⃗ + c⃗| |b⃗| + |c⃗|
Step 3: Conclude that |a⃗| |b⃗| + |c⃗|, with equality only when b⃗ and c⃗ are collinear with same direction