Find the vector vec c collinear with vec a+vec b , given vec…

Find the vector $\vec c$ collinear with $\vec a+\vec b$, given $\vec a\cdot\vec b=5$, $\vec c\cdot\vec b=18$, $|\vec b|=2$.

Explanation of Method

Vector c collinear with a+b from given scalar products.

Step 1: Collinearity

\[ \vec c=\lambda(\vec a+\vec b). \]

Step 2: The condition $\vec c\cdot\vec b=18$

\[ \lambda\left(\vec a\cdot\vec b+|\vec b|^2\right)=\lambda(5+4)=18\ \Longrightarrow\ \lambda=2. \]

\[\vec c=2(\vec a+\vec b)\]

Check: $2(5+4)=18$ $\checkmark$

The solution is unique because $(\vec a+\vec b)\cdot\vec b=9\neq0$.

$O$ $\vec b$ $\vec a$ $\vec a+\vec b$ $\vec c$ $2.5$ $4.5$ $9$

$\vec c=2(\vec a+\vec b)$

Projections onto $\vec b$ ($|\vec b|=2$): $\vec a\cdot\vec b=5\Rightarrow2.5$; $(\vec a+\vec b)\cdot\vec b=9\Rightarrow4.5$; $\vec c\cdot\vec b=18\Rightarrow9$. So $\vec c$ must be twice $\vec a+\vec b$. (Any $\vec a$ with $\vec a\cdot\vec b=5$ gives the same picture.)

Result

\[\boxed{\,\vec c=2(\vec a+\vec b)\,}\]