If vec{a} and vec{b} are two collinear vector | vedclass

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(C) If $\vec{a}$ and $\vec{b}$ are two collinear vectors,they are parallel to the same line.$1$. By definition,$\vec{b}=\lambda \vec{a}$ for some scal

Vedclass · Accepted Answer

The respective components of $\vec{a}$ and $\vec{b}$ are not proportional.

Vedclass · Answer

(C) If $\vec{a}$ and $\vec{b}$ are two collinear vectors,they are parallel to the same line.$1$. By definition,$\vec{b}=\lambda \vec{a}$ for some scalar $\lambda 
eq 0$. Thus,option $A$ is correct.$2$. If $\lambda = \pm 1$,then $\vec{a} = \pm \vec{b}$. This is a specific case of collinearity,so option $B$ is a possible property.$3$. If $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$ and $\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}$,then $\vec{b} = \lambda \vec{a}$ implies $b_1 = \lambda a_1, b_2 = \lambda a_2, b_3 = \lambda a_3$. This means $\frac{b_1}{a_1} = \frac{b_2}{a_2} = \frac{b_3}{a_3} = \lambda$. Thus,the components are proportional. Option $C$ states they are not proportional,which is incorrect.$4$. Collinear vectors can have the same or opposite directions. Option $D$ claims they must have the same direction,which is not necessarily true. However,in the context of multiple-choice questions where only one is 'incorrect' and $C$ is mathematically false,$C$ is the primary incorrect statement regarding properties. Note: $D$ is also technically incorrect as they can have opposite directions. Given the standard interpretation,$C$ is the most fundamentally incorrect statement regarding the definition of collinearity.

If $\vec{a}$ and $\vec{b}$ are two collinear vectors,then which of the following is incorrect?

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