Three points whose position vectors are a + b | vedclass

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(D) Let the points be $A$,$B$,and $C$ with position vectors $\vec{OA} = a + b$,$\vec{OB} = a - b$,and $\vec{OC} = a + kb$.For the points to be colline

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Every real number

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(D) Let the points be $A$,$B$,and $C$ with position vectors $\vec{OA} = a + b$,$\vec{OB} = a - b$,and $\vec{OC} = a + kb$.For the points to be collinear,the vectors $\vec{AB}$ and $\vec{BC}$ must be parallel,meaning $\vec{AB} = \lambda \vec{BC}$ for some scalar $\lambda$.Calculate $\vec{AB} = \vec{OB} - \vec{OA} = (a - b) - (a + b) = -2b$.Calculate $\vec{BC} = \vec{OC} - \vec{OB} = (a + kb) - (a - b) = (k + 1)b$.For collinearity,$-2b = \lambda(k + 1)b$.This implies $-2 = \lambda(k + 1)$. Since $\lambda$ can be any non-zero real number,$k$ can take any real value except for the case where the points coincide. However,in the context of vector collinearity,the condition holds for any $k \in \mathbb{R}$ as long as the vectors are linearly dependent.Thus,the points are collinear for every real number $k$.

Three points whose position vectors are $a + b$,$a - b$,and $a + kb$ will be collinear if the value of $k$ is:

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