Let L be the line \(\vec{r} = \vec{a}+\lambda\vec{b}\), where \(\vec{a}\) and \(\vec{b}\) are non-zero vectors that are not parallel. Which of the following equations also represent the line L?
(A) \(\vec{r} = \vec{a}+\mu(3\vec{b})\)
(B) \(\vec{r} = (\vec{a}+2\vec{b})+\mu\vec{b}\)
(C) \(\vec{r} = 2\vec{a}+\mu\vec{b}\)
(D) \(\vec{r} = \vec{a}+\mu(\vec{a}+\vec{b})\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (B) only
An equation \(\vec{r} = \vec{c}+\mu\vec{d}\) represents the same line L exactly when \(\vec{d}\) is parallel to \(\vec{b}\) and the point \(\vec{c}\) lies on L.
A point \(\vec{c}\) lies on L when \(\vec{c} = \vec{a}+\lambda\vec{b}\) for some \(\lambda\). For \(\vec{c} = 2\vec{a}\) this needs
\(2\vec{a} = \vec{a}+\lambda\vec{b}\)
\(\vec{a} = \lambda\vec{b}\)
which is impossible, because \(\vec{a}\) and \(\vec{b}\) are not parallel. For the same reason \(\vec{a}+\vec{b}\) is not parallel to \(\vec{b}\).
So \(3\vec{b}\) is parallel to \(\vec{b}\), \(\vec{a}+2\vec{b}\) is a point of L, \(2\vec{a}\) is not a point of L, and \(\vec{a}+\vec{b}\) is not parallel to \(\vec{b}\).
Checking the statements
(A) \(\vec{r} = \vec{a}+\mu(3\vec{b})\) — Correct. Same point \(\vec{a}\), and \(3\vec{b}\) is parallel to \(\vec{b}\).
(B) \(\vec{r} = (\vec{a}+2\vec{b})+\mu\vec{b}\) — Correct. \(\vec{a}+2\vec{b}\) is the point \(\lambda = 2\) of L, and the direction is \(\vec{b}\).
(C) \(\vec{r} = 2\vec{a}+\mu\vec{b}\) — Incorrect. \(2\vec{a}\) is on L only if \(\vec{a} = \lambda\vec{b}\); this is a parallel line, not L.
(D) \(\vec{r} = \vec{a}+\mu(\vec{a}+\vec{b})\) — Incorrect. \(\vec{a}+\vec{b}\) is not parallel to \(\vec{b}\), so this line meets L only at \(\vec{a}\).