Question

# \displaystyle=\sqrt{{{3}{i}}}+\pi{j}+{c}{k},{v}={4}{i}−{j}−{k} where is a constant.

Trigonometric Functions

$$\displaystyle=\sqrt{{{3}{i}}}+\pi{j}+{c}{k},{v}={4}{i}−{j}−{k}$$, where is a constant.
(a) Compute $$‖u‖$$, $$‖v‖$$, and $$u \times v$$ for the given vector in R3.
(b) Verify the Cauchy-Schwarz inequality for the given pair of vector.

2021-02-03

Here $$u=\sqrt{3i}+\pi j+ck,\ v=4i−j−k$$, where c is a constant.

(a) $$\displaystyle{\left|{\left|{u}\right|}\right|}=\sqrt{{\sqrt{{3}}^{{{2}}}+π^{{{2}}}+{c}^{{{2}}}=\sqrt{{{3}+π^{{{2}}}+{c}^{{{2}}}}}}}$$
$$\displaystyle{\left|{\left|{v}\right|}\right|}=\sqrt{{{4}^{{{2}}}+{1}+{1}}}=\sqrt{{{18}}}$$
and
$$\displaystyle{u}\cdot{v}={\left(\sqrt{{3}}{i}+π{j}+{c}{k}\right)}\cdot{\left({4}{i}−{j}−{k}\right)}={4}\sqrt{{3}}-π-{c}.$$
(b) $$\displaystyle{\left|{u}\cdot{v}\right|}={\left|{4}\sqrt{{3}}-π-{c}\right|}\leq\sqrt{{{3}+π^{{{2}}}+{c}^{{{2}}}}}\sqrt{{18}}={\left|{\left|{u}\right|}\right|}{\left|{\left|{v}\right|}\right|}.$$