91Ó°ÊÓ

Three vectors are given as:

$\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{1}}}\mathbf{=}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}$

role="math" localid="1654741486353" $\stackrel{\mathbf{→}}{{\mathbf{d}}_2}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{4}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}$

$\stackrel{\mathbf{→}}{{\mathbf{d}}_3}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}$

This problem refers to vector operations that include vector addition, dot product and cross product. Dot product is a scalar quantity whereas cross product is a vector quantity.

The expression for the vector product is given as follows:

$\left(\stackrel{→}{a}×\stackrel{→}{b}\right)\mathbf{}\mathbf{=}\left(a_y{b}_z-b_y{a}_z\right)\hat{\mathbf{i}}\mathbf{+}\left(a_z{b}_x-b_z{a}_x\right)\hat{\mathbf{j}}\mathbf{+}\left(a_x{b}_y-b_y{a}_x\right)\hat{\mathbf{k}}$ … (i)

The expression for the dot product is given as follows:

$\left(\stackrel{→}{a}.\stackrel{→}{b}\right)\mathbf{}\mathbf{=}\left(a_x{b}_x+a_y{b}_y+a_z{b}_z\right)$ … (ii)

First, addition of ${}^{\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}}$and $\stackrel{\mathbf{→}}{{}^{{\mathbf{d}}_{\mathbf{3}}}}$gives,

role="math" localid="1654743006461" $$\begin{gathered} \mathbf{(}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{+}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{)}\hat{\mathbf{j}}\mathbf{+}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{)}\hat{\mathbf{k}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{k}} \end{gathered}$$

Now, the dot product of $\stackrel{\mathbf{→}}{{}^{{\mathbf{d}}_{\mathbf{1}}}}$gives,

$$\begin{gathered} \stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{1}}}\mathbf{.}\mathbf{(}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{+}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)}\mathbf{}\mathbf{.}\mathbf{}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{+}\mathbf{6}\mathbf{.}\mathbf{0} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}{\mathbf{m}}^{\mathbf{2}} \end{gathered}$$

The cross product of ${}^{\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}}$and ${}^{\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}}$gives,

$\mathbf{(}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{×}\mathbf{}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{-}\left(-2.0\right)\mathbf{)}\hat{\mathbf{j}}\mathbf{+}\mathbf{(}\left(-6.0\right)\mathbf{-}\left(-8.0\right)\mathbf{)}\hat{\mathbf{k}}$

$\mathbf{(}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{×}\mathbf{}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{10}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{6}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}$

Now, the dot product of $\stackrel{\mathbf{→}}{{}^{{\mathbf{d}}_{\mathbf{1}}}}$gives,

role="math" localid="1654743321424" $$\begin{gathered} \stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{1}}}\mathbf{.}\mathbf{(}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{×}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)}\mathbf{}\mathbf{.}\mathbf{}\mathbf{(}\mathbf{-}\mathbf{10}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{6}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{+}\mathbf{18}\mathbf{.}\mathbf{0}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{0} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{52}\mathbf{.}\mathbf{0}\mathbf{}{\mathbf{m}}^3 \end{gathered}$$

The addition of ${}^{\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}}$and $\stackrel{\mathbf{→}}{{}^{{\mathbf{d}}_{\mathbf{3}}}}$gives ,

$\mathbf{(}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{+}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}$

Now, now the cross product of ${}^{\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{1}}}}$gives,

localid="1660891291873" $$\begin{gathered} \stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{1}}}×\mathbf{(}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{2}}}\mathbf{}\mathbf{+}\stackrel{\mathbf{→}}{{\mathbf{d}}_{\mathbf{3}}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)}\mathbf{}\mathbf{×}\mathbf{}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\hat{\mathbf{k}}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{(}\mathbf{11}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{)}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{9}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{)}\hat{\mathbf{j}}\mathbf{+}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{)}\hat{\mathbf{k}} \end{gathered}$$