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-A- Both 1 and 2 are falseB- 1 is true and 2 is falseC- Both 1 and 2 are trueD- 2 is true and 1 is false

The correct option is C Both 1 and 2 are trueTwo vectors a and b are linearly dependent if for x, y ϵ R : xa +yb=0 where x and y are not simultaneously zero ...

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Byju's Answer

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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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.