Question

Which of the following statements is / are true?
1.Set of two vectors $\overrightarrow{a},\overrightarrow{b}$ is linearly dependent if and only if either any of $\overrightarrow{a}\text{and}\overrightarrow{b}$ is zero or they are parallel
2.$\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}$ are linearly dependent $\iff \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}$ are coplanar.

• A. Both $1$ and $2$ are false
• B. $1$ is true and $2$ is false
• C. Both $1$ and $2$ are true
• D. $2$ is true and $1$ is false

Answer 1

The correct option is C Both $1$ and $2$ are true

Two vectors $\overrightarrow{a}\text{and}\overrightarrow{b}$ are linearly dependent if for $x,yϵR:x\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{0}$ where $x$ and $y$ are not simultaneously zero.
For linear dependence following cases are possible
$x=0,y\ne 0$ $\Rightarrow \overrightarrow{b}=0$
$x\ne 0,y=0$ $\Rightarrow \overrightarrow{a}=0$
$x\text{and}y\ne 0$ The case where the vectors are parallel.
So $x\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{0}$ if either $\overrightarrow{a}\text{and}\overrightarrow{b}$ and parallel or one of them is zero.
Statement 2:
In the statement it is given that $\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}$ are linearly dependent. That means one of them can be written as a linear combination of other two.
$\text{So}x\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{c}$
From fundamental theorem in 2D, we know that any linear combination of two vectors, gives a vector the same plane as that of $\overrightarrow{a}\text{and}\overrightarrow{b}$.
$x\overrightarrow{a}+y\overrightarrow{b}$ is coplanar with $\overrightarrow{a}\text{and}\overrightarrow{b}$
Let's say $x\overrightarrow{a}+y\overrightarrow{b}=\overrightarrow{c}$
So $\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}$ are coplanar