# Recent questions tagged euclidean

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Find the values of $w,x,y,z$
Find the values of $w,x,y,z$Find the values of $w,x,y,z$ \ \begin{aligned} &amp;w+x+y=49 \\ &amp;x+y+z=30 \\ &amp;y+z+w=35 \\ &amp;z+w+x=42 \end{aligned} \ ...
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What is the dot (Euclidean inner) product of two vectors?
What is the dot (Euclidean inner) product of two vectors?What is the dot (Euclidean inner) product of two vectors? ...
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What is the component form of the dot product?
What is the component form of the dot product?What is the component form of the dot product? ...
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Which mathematician came up with the dot product notation?
Which mathematician came up with the dot product notation?Which mathematician came up with the dot product notation? ...
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Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$
Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$ ...
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Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$\mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}$$Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$\mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\| ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Proof or counterexample. Here $v, w, z$ are vectors in a real inner product space $H$. 0 answers 229 views Proof or counterexample. Here $v, w, z$ are vectors in a real inner product space $H$.Proof or counterexample. Here $v, w, z$ are vectors in a real inner product space $H$. a) Let $v, w, z$ be vectors in a real inner product space ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $U, V, W$ be orthogonal vectors and let $Z=a U+b V+c W$, where $a, b, c$ are scalars. 0 answers 57 views 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 Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find all vectors in the plane (through the origin) spanned by \(\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector $\mathbf{Z}=(2,1,2)$. 0 answers 188 views Find all vectors in the plane (through the origin) spanned by $\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector $\mathbf{Z}=(2,1,2)$.Find all vectors in the plane (through the origin) spanned by $\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector ... close 0 answers 230 views Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and $A \vec{w}$ are orthogonal.Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and ... close 0 answers 291 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 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Answer the following in terms of \(\mathbf{V}, \mathbf{W}$, and $\mathbf{Z}$. 0 answers 70 views Answer the following in terms of $\mathbf{V}, \mathbf{W}$, and $\mathbf{Z}$.Let $A$ be a matrix, not necessarily square. Say $\mathbf{V}$ and $\mathbf{W}$ are particular solutions of the equations $A \mathbf{V}=\mathbf{ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy-Schwarz inequality holds. \(\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$ 0 answers 66 views Verify that the Cauchy-Schwarz inequality holds. $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$Verify that the Cauchy-Schwarz inequality holds. (b) $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1) 0 answers 51 views Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)Verify that the Cauchy&amp;ndash;Schwarz inequality holds for \mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1) ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3) 0 answers 65 views Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)Verify that the Cauchy&amp;ndash;Schwarz inequality holds for \mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3) ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2) 0 answers 48 views Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)Verify that the Cauchy&amp;ndash;Schwarz inequality holds for \mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2) ... close 0 answers 292 views Let \mathbf{r}_{0}=\left(x_{0}, y_{0}\right) be a fixed vector in R^{2}. In each part, describe in words the set of all vectors \mathbf{r}=(x, y) that satisfy the stated condition.Let \mathbf{r}_{0}=\left(x_{0}, y_{0}\right) be a fixed vector in R^{2}. In each part, describe in words the set of all vectors \mathbf{r}=(x, y) ... 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 0 answers 55 views 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$$ \cos \ ...
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Let $\mathbf{u}$ be a vector in $R^{100}$ whose $i$ th component is $i$, and let $\mathbf{v}$ be the vector in $R^{100}$ whose $i$ th component is $1 /(i+1)$. Find the dot product of $\mathbf{u}$ and $\mathbf{v}$.Let $\mathbf{u}$ be a vector in $R^{100}$ whose $i$ th component is $i$, and let $\mathbf{v}$ be the vector in $R^{100}$ whose $i$ th component is $1 ... close 1 answer 324 views If$\mathbf{u}$and$\mathbf{v}$are orthogonal vectors in$R^{n}$with the Euclidean inner product, then prove that $$\|\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{u}=(-2,3,1,4) and \mathbf{v}=(1,2,0,-1) are orthogonal 1 answer 196 views Show that \mathbf{u}=(-2,3,1,4) and \mathbf{v}=(1,2,0,-1) are orthogonalShow that \mathbf{u}=(-2,3,1,4) and \mathbf{v}=(1,2,0,-1) are orthogonal ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the distance D between the point P_{0}\left(x_{0}, y_{0},z_{0}\right) and the line a x+b y+c z+d=0 in R^{3} ? 1 answer 54 views What is the distance D between the point P_{0}\left(x_{0}, y_{0},z_{0}\right) and the line a x+b y+c z+d=0 in R^{3} ?What is the distance D between the point P_{0}\left(x_{0}, y_{0},z_{0}\right) and the line a x+b y+c z+d=0 in R^{3} ? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Prove that, In R^{3} the distance D between the point P_{0}\left(x_{0}, y_{0}, z_{0}\right) and the plane a x+b y+c z+d=0 is 1 answer 218 views Prove that, In R^{3} the distance D between the point P_{0}\left(x_{0}, y_{0}, z_{0}\right) and the plane a x+b y+c z+d=0 isProve that, In R^{3} the distance D between the point P_{0}\left(x_{0}, y_{0}, z_{0}\right) and the plane a x+b y+c z+d=0 is$$ D=\frac{\left| ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find the distance$D$between the point$(1,-4,-3)$and the plane$2 x-3 y+6 z=-1$. 1 answer 50 views Find the distance$D$between the point$(1,-4,-3)$and the plane$2 x-3 y+6 z=-1$.Find the distance$D$between the point$(1,-4,-3)$and the plane$2 x-3 y+6 z=-1$. ... close 1 answer 173 views The planes $$x+2 y-2 z=3 \text { and } 2 x+4 y-4 z=7$$ are parallel since their normals,$(1,2,-2)$and$(2,4,-4)$, are parallel vectors. Find the distance between these planes.The planes $$x+2 y-2 z=3 \text { and } 2 x+4 y-4 z=7$$ are parallel since their normals,$(1,2,-2)$and$(2,4,-4)$, are parallel vectors. Find the d ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that$\mathbf{v}=(a, b)$and$\mathbf{w}=(-b, a)$are orthogonal vectors. 0 answers 232 views Show that$\mathbf{v}=(a, b)$and$\mathbf{w}=(-b, a)$are orthogonal vectors.Show that$\mathbf{v}=(a, b)$and$\mathbf{w}=(-b, a)$are orthogonal vectors. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Do the points$A(1,1,1), B(-2,0,3)$, and$C(-3,-1,1)$form the vertices of a right triangle? Explain. 0 answers 151 views Do the points$A(1,1,1), B(-2,0,3)$, and$C(-3,-1,1)$form the vertices of a right triangle? Explain.Do the points$A(1,1,1), B(-2,0,3)$, and$C(-3,-1,1)$form the vertices of a right triangle? Explain. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 A sailboat travels$100 \mathrm{~m}$due north while the wind exerts a force of$500 \mathrm{~N}$toward the northeast. How much work does the wind do? 0 answers 45 views A sailboat travels$100 \mathrm{~m}$due north while the wind exerts a force of$500 \mathrm{~N}$toward the northeast. How much work does the wind do?A sailboat travels$100 \mathrm{~m}$due north while the wind exerts a force of$500 \mathrm{~N}$toward the northeast. How much work does the wind do ... close 0 answers 289 views close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find the lengths of the sides and the interior angles of the triangle in$R^{4}$whose vertices are $$P(2,4,2,4,2), \quad Q(6,4,4,4,6), \quad R(5,7,5,7,2)$$ 0 answers 44 views Find the lengths of the sides and the interior angles of the triangle in$R^{4}$whose vertices are $$P(2,4,2,4,2), \quad Q(6,4,4,4,6), \quad R(5,7,5,7,2)$$Find the lengths of the sides and the interior angles of the triangle in$R^{4}$whose vertices are $$P(2,4,2,4,2), \quad Q(6,4,4,4,6), \quad R(5,7,5 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 The angle between vectors (1,3,-2) and (4,-2,-1) is 0 answers 155 views The angle between vectors (1,3,-2) and (4,-2,-1) isThe angle between vectors (1,3,-2) and (4,-2,-1) is &nbsp; A) 0 B) \dfrac{\pi}{3} C) \dfrac{\pi}{2} D) \pi E) none of the above ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 The distance from the point (1,1,1) to the plane 2 x-10 y+11 z-4=0 is equal to 0 answers 54 views The distance from the point (1,1,1) to the plane 2 x-10 y+11 z-4=0 is equal toThe distance from the point (1,1,1) to the plane 2 x-10 y+11 z-4=0 is equal to &nbsp; A) \dfrac{1}{3} B) 3 C) \dfrac{1}{15} D) 5 E) no ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Determine whether planes 2 x+y+z-1=0,-x+3 y-2 z-3=0 and 3 x-y=-1 intersect. If yes, find the intersection. 0 answers 151 views Determine whether planes 2 x+y+z-1=0,-x+3 y-2 z-3=0 and 3 x-y=-1 intersect. If yes, find the intersection.Determine whether planes 2 x+y+z-1=0,-x+3 y-2 z-3=0 and 3 x-y=-1 intersect. If yes, find the intersection. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the distance D between the point P_{0}\left(x_{0}, y_{0}\right) and the line a x+b y+c=0 in R^2? 1 answer 208 views What is the distance D between the point P_{0}\left(x_{0}, y_{0}\right) and the line a x+b y+c=0 in R^2?What is the distance D between the point P_{0}\left(x_{0}, y_{0}\right) and the line a x+b y+c=0 in R^2? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 The angle between vectors (2,6,-4) and (4,-2,-1) is 0 answers 146 views The angle between vectors (2,6,-4) and (4,-2,-1) isThe angle between vectors (2,6,-4) and (4,-2,-1) is &nbsp; A. 0 B. \dfrac{\pi}{3} C. \dfrac{\pi}{2} D. \pi E. none of the above ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 The angle between vectors (1,3,-2) and (4,-2,-1) is 0 answers 145 views The angle between vectors (1,3,-2) and (4,-2,-1) isThe angle between vectors (1,3,-2) and (4,-2,-1) is &nbsp; A) 0 B) \frac{\pi}{3} C) \frac{\pi}{2} D) \pi E) none of the above ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 The distance from the point (-1,1,1) to the plane 0 answers 39 views The distance from the point (-1,1,1) to the planeThe distance from the point (-1,1,1) to the plane$$ 2 x-10 y+11 z-4=0$$is equal to &nbsp; A) 3 B)$\frac{1}{3}$C) 5 D)$\frac{1}{5}\$ E) no ...
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When are bases in Euclidean vector spaces V (Rⁿ or Cⁿ) topologically stable?
When are bases in Euclidean vector spaces V (Rⁿ or Cⁿ) topologically stable?When are bases in Euclidean vector spaces V (Rⁿ or Cⁿ) &nbsp;topologically stable? ...
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What are normed spaces?
What are normed spaces?What are normed spaces? ...
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How do I describe a Euclidean space?
How do I describe a Euclidean space?How do I describe a Euclidean space? ...
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Calculate the Euclidean distance between the points P(-2;-1) and Q(2;-6)
Calculate the Euclidean distance between the points P(-2;-1) and Q(2;-6)Calculate the Euclidean distance between the points P(-2;-1) and Q(2;-6) ...
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What is a key step in K-nearest neighbors?
What is a key step in K-nearest neighbors?What is a key step in K-nearest neighbors? ...
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How do I choose k when using k-nearest neighbor algorithm?
How do I choose k when using k-nearest neighbor algorithm?How do I choose k when using k-nearest neighbor algorithm? ...
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In k-NN what will happen when you increase/decrease the value of k?
In k-NN what will happen when you increase/decrease the value of k?In k-NN what will happen when you increase/decrease the value of k? ...
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How does one choose between Euclidean and Manhattan distances when using k-NN algorithm?