# Chapter 12: Vectors and the Geometry of Space - Section 12.3 - The Dot Product - Exercises 12.3 - Page 713: 27

$a$

#### Work Step by Step

$(*)\quad {\bf u_{1}}$and ${\bf u_{2}}$ are orthogonal $\Rightarrow{\bf u_{1}}\cdot{\bf u_{2}}=0$ $(**) \quad {\bf u_{1}}$and ${\bf u_{2}}$ are unit vectors $\Rightarrow|{\bf u_{1}}|=|{\bf u_{2}}|=1$ Using Properties of the Dot Product (boxed on p. 709): ${\bf v}\cdot{\bf u_{1}}=(a{\bf u_{1}}+b{\bf u_{2}})\cdot{\bf u_{1}}$ $=a{\bf u_{1}} \cdot{\bf u_{1}}+b{\bf u_{2}}\cdot{\bf u_{1}} \qquad$... property 3 $=a({\bf u_{1}} \cdot{\bf u_{1}})+b({\bf u_{2}}\cdot{\bf u_{1}}) \qquad$... property 2 $=a({\bf u_{1}} \cdot{\bf u_{1}})+b(0) \qquad$... $(*)$ from above $=a\cdot|{\bf u_{1}}|^{2} \qquad$... property $4$ $=a(1) \qquad$... $(**)$ from above $=a$

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