Chapter 11 - Vectors and the Geometry of Space - 11.3 Exercises - Page 773: 2

$$\eqalign{ & {\text{Summary}} \cr & \left( {\bf{a}} \right)22 \cr & \left( {\bf{b}} \right)116 \cr & \left( {\bf{c}} \right)116 \cr & \left( {\bf{d}} \right)\left\langle { - 44,66} \right\rangle \cr & \left( {\bf{e}} \right)16 \cr} $$

$$\eqalign{ & {\bf{u}} = \left\langle {4,10} \right\rangle ,{\text{ }}{\bf{v}} = \left\langle { - 2,3} \right\rangle \cr & \cr & \left( {\bf{a}} \right){\text{Find }}{\bf{u}} \cdot {\bf{v}} \cr & {\bf{u}} \cdot {\bf{v}} = \left\langle {4,10} \right\rangle \cdot \left\langle { - 2,3} \right\rangle \cr & {\bf{u}} \cdot {\bf{v}} = - 8 + 30 \cr & {\bf{u}} \cdot {\bf{v}} = 22 \cr & \cr & \left( {\bf{b}} \right){\text{Find }}{\bf{u}} \cdot {\bf{u}} \cr & {\bf{u}} \cdot {\bf{u}} = \left\langle {4,10} \right\rangle \cdot \left\langle {4,10} \right\rangle \cr & {\bf{u}} \cdot {\bf{u}} = 16 + 100 \cr & {\bf{u}} \cdot {\bf{u}} = 116 \cr & \cr & \left( {\bf{c}} \right){\text{Find }}{\left\| {\bf{u}} \right\|^2} \cr & {\left\| {\bf{u}} \right\|^2} = {\left\| {\left\langle {4,10} \right\rangle } \right\|^2} \cr & {\left\| {\bf{u}} \right\|^2} = {\left( {\sqrt {{{\left( 4 \right)}^2} + {{\left( {10} \right)}^2}} } \right)^2} \cr & {\left\| {\bf{u}} \right\|^2} = {\left( {\sqrt {116} } \right)^2} \cr & {\left\| {\bf{u}} \right\|^2} = 116 \cr & \cr & \left( {\bf{d}} \right){\text{Find }}\left( {{\bf{u}} \cdot {\bf{v}}} \right){\bf{v}} \cr & \left( {{\bf{u}} \cdot {\bf{v}}} \right){\bf{v}} = 22\left\langle { - 2,3} \right\rangle \cr & \left( {{\bf{u}} \cdot {\bf{v}}} \right){\bf{v}} = \left\langle { - 44,66} \right\rangle \cr & \cr & \left( {\bf{e}} \right){\text{Find }}{\bf{u}} \cdot \left( {2{\bf{v}}} \right) \cr & {\bf{u}} \cdot \left( {2{\bf{v}}} \right) = \left\langle {4,10} \right\rangle \cdot \left( {2\left\langle { - 2,3} \right\rangle } \right) \cr & {\bf{u}} \cdot \left( {2{\bf{v}}} \right) = \left\langle {4,10} \right\rangle \cdot \left\langle { - 4,6} \right\rangle \cr & {\bf{u}} \cdot \left( {2{\bf{v}}} \right) = - 44 + 60 \cr & {\bf{u}} \cdot \left( {2{\bf{v}}} \right) = 16 \cr} $$