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Question
It is found that |A + B| = |A|.This necessarily implies ______.
Options
B = 0
A, B are antiparallel
A, B are perpendicular
A . B ≤ 0
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Solution
It is found that |A + B| = |A|.This necessarily implies A, B are antiparallel.
Explanation:
According to the problem,
`|vecA + vecB| = |vecA|`
By squaring both sides, we get
`|vecA + vecB|^2 = |vecA|^2`
⇒ `|vecA|^2 + |vecB|^2 + 2|vecA||vecB| cos theta = |vecA|^2`
Where θ is the angle between `vecA` and `vecB`
`|vecB| (|vecB| + 2|vecA| cos theta)` = 0
⇒ `|vecB| + 2|vecA| cos theta` = 0
⇒ `cos theta = (|vecB|)/(2|vecA|)`
If A and B are antiparallel, then θ = 180°
Hence from equation (i),
`- 1 = (|vecB|)/(2|A|)` ⇒ `|vecB| = 2|vecA|`
Hence, correct answer will be `vecA` and `vecB` are antiparallel provided `|vecB| = 2|vecA|`.
It seem like option (a) is also correct but it is not for `|vecA + vecB| = |vecA|`, either `vecB` = 0 or `vecB = - 2 vecA`, so this option will be false.
