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Fundamental Theorem in 2D

Two vectors ...

Question

Two vectors $¯ ¯¯ ¯ A$ and $¯ ¯¯ ¯ B$ lie in a plane. A third vector $¯ ¯¯ ¯ C$ lies outside this plane. Then the sum of these vectors $¯ ¯¯ ¯ A + ¯ ¯¯ ¯ B + ¯ ¯¯ ¯ C$ :

can be zero

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can never be zero

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lies in plane containing

¯ ¯¯ ¯ A + ¯ ¯¯ ¯ B

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lies in a plane containing

¯ ¯¯ ¯ A - ¯ ¯¯ ¯ B

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Solution

The correct option is B can never be zero
Let the vectors lie in $X - Y$ plane.
We can write the vectors as $\to A = A_{x}^x + A_{y}^y$ and $\to B = B_{x}^x + B_{y}^y$
Vector C lies outside the plane, so $\to C = C_{x}^x + C_{y}^y + C_{z}^z$
We get $\to A + \to B + \to C$ $= (A_{x} + B_{x} + C_{x})^x + (A_{y} + B_{y} + C_{y})^y + C_{z}^z$
We can have some values such that the value of $(A_{x} + B_{x} + C_{x})$ can be zero and $(A_{y} + B_{y} + C_{y})$ can be zero. But $C_{z}$ can not be zero as it is given that $\to C$ lies outside the plane.
$⟹$ $\to A + \to B + \to C \neq 0$
So, option B is correct.

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Two vectors ¯¯¯¯A and ¯¯¯¯B lie in a plane. A third vector ¯¯¯¯C lies outside this plane. Then the sum of these vectors ¯¯¯¯A+¯¯¯¯B+¯¯¯¯C:

Two vectors $¯ ¯¯ ¯ A$ and $¯ ¯¯ ¯ B$ lie in a plane. A third vector $¯ ¯¯ ¯ C$ lies outside this plane. Then the sum of these vectors $¯ ¯¯ ¯ A + ¯ ¯¯ ¯ B + ¯ ¯¯ ¯ C$ :