Question

TRUE OR FALSE?If $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{y}}$are two vectors in${\mathit{R}}^{\mathbf{n}}$, then the equation role="math" localid="1659506190737" $\mathbf{\parallel }\overrightarrow{\mathbf{x}}\mathbf{+}\overrightarrow{\mathbf{y}}{\mathbf{\parallel }}^{\mathbf{2}}\mathbf{=}\mathbf{\parallel }\overrightarrow{\mathbf{x}}{\mathbf{\parallel }}^{\mathbf{2}}\mathbf{+}\mathbf{\parallel }\overrightarrow{\mathbf{y}}{\mathbf{\parallel }}^{\mathbf{2}}$must hold.

Short answer

The given statement is false.

Step-by-step solution

Statement of a Pythagorean Theorem.

Suppose $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{y}}$andare two vectors in${\mathit{R}}^{\mathbf{n}}$.

Then the equationrole="math" localid="1659507544082" $\mathbf{\parallel }\overrightarrow{\mathbf{x}}\mathbf{+}\overrightarrow{\mathbf{y}}{\mathbf{\parallel }}^{\mathbf{2}}\mathbf{=}\mathbf{\parallel }\overrightarrow{\mathbf{x}}{\mathbf{\parallel }}^{\mathbf{2}}\mathbf{+}\mathbf{\parallel }\overrightarrow{\mathbf{y}}{\mathbf{\parallel }}^{\mathbf{2}}$ holds if and only ifandare orthogonal.

Check whether the given statement is a true or false.

The given statement is if $\stackrel{\rightharpoonup }{x}and\stackrel{\rightharpoonup }{y}$and are two vectors in $R^n$, then the equation role="math" localid="1659507570743" $\parallel \overrightarrow{\mathrm{x}}+\overrightarrow{\mathrm{y}}{\parallel }^2=\parallel \overrightarrow{\mathrm{x}}{\parallel }^2+\parallel \overrightarrow{\mathrm{y}}{\parallel }^2$must hold.

By the Pythagorean Theorem, the equation$\parallel \overrightarrow{\mathrm{x}}+\overrightarrow{\mathrm{y}}{\parallel }^2=\parallel \overrightarrow{\mathrm{x}}{\parallel }^2+\parallel \overrightarrow{\mathrm{y}}{\parallel }^2$ is holds only when$\stackrel{\rightharpoonup }{x}$ and $\stackrel{\rightharpoonup }{y}$are orthogonal.

Then the given statement is false.