Let \(\vec{a}\) be a non-zero vector and \(\lambda\) a scalar. Which of the following statements are correct?
(A) \(|\lambda\vec{a}| = |\lambda|\,|\vec{a}|\) for every scalar \(\lambda\)
(B) If \(\lambda \lt 0\), then \(\lambda\vec{a}\) has the same direction as \(\vec{a}\)
(C) \(\displaystyle\frac{\vec{a}}{|\vec{a}|}\) is a unit vector
(D) \(\vec{a}\) and \(\lambda\vec{a}\) are collinear for every non-zero scalar \(\lambda\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
Multiplying \(\vec{a}\) by a scalar \(\lambda\) multiplies its length by \(|\lambda|\). The direction stays the same when \(\lambda \gt 0\) and is reversed when \(\lambda \lt 0\); either way \(\lambda\vec{a}\) lies along the same line as \(\vec{a}\).
So \(|\lambda\vec{a}| = |\lambda|\,|\vec{a}|\), a negative \(\lambda\) reverses the direction, \(\frac{\vec{a}}{|\vec{a}|}\) has length 1, and \(\lambda\vec{a}\) is always along the line of \(\vec{a}\).
Checking the statements
(A) \(|\lambda\vec{a}| = |\lambda|\,|\vec{a}|\) — Correct. The length is multiplied by \(|\lambda|\); for example \(|-2\vec{a}| = 2|\vec{a}|\).
(B) For \(\lambda \lt 0\), \(\lambda\vec{a}\) has the direction of \(\vec{a}\) — Incorrect. A negative scalar gives the opposite direction; \(-\vec{a}\) points opposite to \(\vec{a}\).
(C) \(\frac{\vec{a}}{|\vec{a}|}\) is a unit vector — Correct. Its length is \(\frac{1}{|\vec{a}|}\times|\vec{a}| = 1\).
(D) \(\vec{a}\) and \(\lambda\vec{a}\) are collinear — Correct. \(\lambda\vec{a}\) is parallel to \(\vec{a}\), so the two are collinear.