# arrow_back When are two nonzero vectors orthogonal to each other?

25 views
When are two nonzero vectors orthogonal to each other?

Remember $$\theta=\cos ^{-1}\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\right)$$ It follows from this that $\theta=\pi / 2$ if and only if $\mathbf{u} \cdot \mathbf{v}=0$. Thus, Two nonzero vectors $\mathbf{u}$ and $\mathbf{v}$ in $R^{n}$ are said to be orthogonal (or perpendicular) if $\mathbf{u} \cdot \mathbf{v}=0 .$ We will also agree that the zero vector in $R^{n}$ is orthogonal to every vector in $R^{n}$.
by Platinum
(106,844 points)

## Related questions

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
What are Orthogonal Projections?
What are Orthogonal Projections?What are Orthogonal Projections? ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Show that two nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in $R^{3}$ are orthogonal if and only if their direction cosines satisfy
Show that two nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in $R^{3}$ are orthogonal if and only if their direction cosines satisfyShow that two nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in $R^{3}$ are orthogonal if and only if their direction cosines satisfy  \cos \ ...
Let $U, V, W$ be orthogonal vectors and let $Z=a U+b V+c W$, where $a, b, c$ are scalars.
Let $U, V, W$ be orthogonal vectors and let $Z=a U+b V+c W$, where $a, b, c$ are scalars.Let $U, V, W$ be orthogonal vectors and let $Z=a U+b V+c W$, where $a, b, c$ are scalars. a) (Pythagoras) Show that $\|Z\|^{2}=a^{2}\|U\|^{2}+b ... close 1 answer 12 views If \mathbf{u} and \mathbf{v} are orthogonal vectors in R^{n} with the Euclidean inner product, then prove that  \|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2} If \mathbf{u} and \mathbf{v} are orthogonal vectors in R^{n} with the Euclidean inner product, then prove that  \|\mathbf{u}+\mathbf{v}\|^{2}= ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that \mathbf{v}=(a, b) and \mathbf{w}=(-b, a) are orthogonal vectors. 0 answers 9 views Show that \mathbf{v}=(a, b) and \mathbf{w}=(-b, a) are orthogonal vectors.Show that \mathbf{v}=(a, b) and \mathbf{w}=(-b, a) are orthogonal vectors. ... close 0 answers 11 views close 0 answers 15 views Let \(\vec{v}$ and $\vec{w}$ be vectors in $\mathbb{R}^{n}$. If $\|\vec{v}\|=\|\vec{w}\|$, show there is an orthogonal matrix $R$ with $R \vec{v}=\vec{w}$ and $R \vec{w}=\vec{v}$.Let $\vec{v}$ and $\vec{w}$ be vectors in $\mathbb{R}^{n}$. If $\|\vec{v}\|=\|\vec{w}\|$, show there is an orthogonal matrix $R$ with \(R \v ...