Which of the following statements is - are true-1-Set of two vectors a-b is linearly dependent if and only if either any of a and b is zero or they are parallel2-a-b and c are linearly dependent -a-b and c are coplanar-

Two vectors a and b are linearly dependent if for x, y ϵ R : xa +yb=0 where x and y are not simultaneously zero. For linear dependence following cases are po ...

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Standard XII

Mathematics

Fundamental Theorem in 2D

Which of the ...

Question

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

Both

1

and

2

are false

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1

is true and

2

is false

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Both

1

and

2

are true

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2

is true and

1

is false

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Solution

The correct option is C Both $1$ and $2$ are true
Two vectors $\to a and \to b$ are linearly dependent if for $x, y ϵ R : x \to a + y \to b = \to 0$ where $x$ and $y$ are not simultaneously zero.
For linear dependence following cases are possible
$x = 0, y \neq 0$ $\Rightarrow \to b = 0$
$x \neq 0, y = 0$ $\Rightarrow \to a = 0$
$x and y \neq 0$ The case where the vectors are parallel.
So $x \to a + y \to b = \to 0$ if either $\to a and \to b$ and parallel or one of them is zero.
Statement 2:
In the statement it is given that $\to a, \to b and \to c$ are linearly dependent. That means one of them can be written as a linear combination of other two.
$So x \to a + y \to b = \to c$
From fundamental theorem in 2D, we know that any linear combination of two vectors, gives a vector the same plane as that of $\to a and \to b$ .
$x \to a + y \to b$ is coplanar with $\to a and \to b$
Let's say $x \to a + y \to b = \to c$
So $\to a, \to b and \to c$ are coplanar

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Similar questions

Q. Which of the following statements is / are true?
1.Set of two vectors

\to a, \to b

is linearly dependent if and only if either any of

\to a and \to b

is zero or they are parallel
2.

\to a, \to b and \to c

are linearly dependent

\Leftrightarrow \to a, \to b and \to c

are coplanar.

Q. Assertion :If

\to a =^i, \to b =^j

and

\to c =^i +^j

then

\to a

and

\to b

are linearly independent but

\to a, \to b

and

\to c

are linearly dependent. Reason: If

\to a

and

\to b

are linearly dependent and

\to c

is any vector then

\to a, \to b

and

\to c

are linearly dependent.

Q. Assertion :STATEMENT-

1

[\to b \times \to c \to c \times \to a \to a \times \to b] = 0

Reason: STATEMENT-

2

: If

\to p, \to q, \to r

are linearly dependent vectors then they are coplanar

Q. If the vectors

\to a + 1343 \to b + \to c, - \to a + \to b + \to c

and

\to a + μ \to b + 2 \to c

are linearly dependent, then

μ =

Q. If the vectors

\to a + \to b + \to c, \to a + λ \to b + 2 \to c

and

- \to a + \to b + \to c

are linearly dependent, then the value of

λ

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Which of the following statements is / are true? 1.Set of two vectors →a,→b is linearly dependent if and only if either any of →a and →b is zero or they are parallel 2.→a,→b and →c are linearly dependent ⇔→a,→b and →c are coplanar.

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