If the points a + b,a - b,and a + kb are coll | vedclass

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(D) Let the points be $P = a + b$,$Q = a - b$,and $R = a + kb$.For the points to be collinear,the vectors $\overrightarrow{PQ}$ and $\overrightarrow{Q

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Any real number

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(D) Let the points be $P = a + b$,$Q = a - b$,and $R = a + kb$.For the points to be collinear,the vectors $\overrightarrow{PQ}$ and $\overrightarrow{QR}$ must be parallel.$\overrightarrow{PQ} = Q - P = (a - b) - (a + b) = -2b$.$\overrightarrow{QR} = R - Q = (a + kb) - (a - b) = (k + 1)b$.For these vectors to be parallel,there must exist a scalar $\lambda$ such that $\overrightarrow{PQ} = \lambda \overrightarrow{QR}$.$-2b = \lambda(k + 1)b$.Since $b$ is a vector,this implies $-2 = \lambda(k + 1)$.This equation holds for any $k$ as long as we can choose a suitable $\lambda$. Specifically,if $k = -1$,the points $Q$ and $R$ coincide,making them collinear. For any other $k$,the vectors are proportional. Thus,$k$ can be any real number.

If the points $a + b$,$a - b$,and $a + kb$ are collinear,then $k =$

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