Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given

Linear Algebra Problems and Solutions

Problem 254

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are
\[\|\mathbf{a}\|=\|\mathbf{b}\|=1\] and the inner product
\[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\]

Then determine the length $\|\mathbf{a}-\mathbf{b}\|$.
(Note that this length is the distance between $\mathbf{a}$ and $\mathbf{b}$.)

 
LoadingAdd to solve later

Sponsored Links

Solution.

Recall that the length of a vector $\mathbf{x}$ is defined to be
\[\|\mathbf{x}\|=\sqrt{\mathbf{x}^{\trans}\mathbf{x}},\] where $\mathbf{x}^{\trans}$ is the transpose of $\mathbf{x}$.

Also, recall that the inner product of two vectors $\mathbf{x}, \mathbf{y}$ are commutative.
Namely we have
\[\mathbf{x}\cdot \mathbf{y}=\mathbf{x}^{\trans}\mathbf{y}=\mathbf{y}^{\trans}\mathbf{x}=\mathbf{y} \cdot \mathbf{x}.\]


Applying the second fact with given vectors $\mathbf{a}, \mathbf{b}$, we obtain
\[\mathbf{a}^{\trans}\mathbf{b}=\mathbf{b}^{\trans}\mathbf{a}= -\frac{1}{2}.\]


Now we compute $\|\mathbf{a}-\mathbf{b}\|^2$ as follows.
We have
\begin{align*}
\|\mathbf{a}-\mathbf{b}\|^2&=(\mathbf{a}-\mathbf{b})^{\trans}(\mathbf{a}-\mathbf{b}) \qquad \text{ (by definition of the length)}\\
&=(\mathbf{a}^{\trans}-\mathbf{b}^{\trans})(\mathbf{a}-\mathbf{b})\\
&=\mathbf{a}^{\trans}\mathbf{a}-\mathbf{a}^{\trans}\mathbf{b}-\mathbf{b}^{\trans}\mathbf{a}+\mathbf{b}^{\trans}\mathbf{b}\\
&=\|\mathbf{a}\|^2-\mathbf{a}^{\trans}\mathbf{b}-\mathbf{b}^{\trans}\mathbf{a}+\|\mathbf{b}\|^2\\
&=1-\left(-\frac{1}{2} \right)-\left(-\frac{1}{2} \right)+1\\
&=3.
\end{align*}

Since the length is nonnegative, we take the square root of the above equality and obtain
\[\|\mathbf{a}-\mathbf{b}\|=\sqrt{3}.\]


LoadingAdd to solve later

Sponsored Links

More from my site

  • Find the Inverse Matrix of a Matrix With FractionsFind the Inverse Matrix of a Matrix With Fractions Find the inverse matrix of the matrix \[A=\begin{bmatrix} \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt] \frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt] -\frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{bmatrix}.\]   Hint. You may use the augmented matrix […]
  • Inner Products, Lengths, and Distances of 3-Dimensional Real VectorsInner Products, Lengths, and Distances of 3-Dimensional Real Vectors For this problem, use the real vectors \[ \mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} . \] Suppose that $\mathbf{v}_4$ is another vector which is […]
  • Inner Product, Norm, and Orthogonal VectorsInner Product, Norm, and Orthogonal Vectors Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in […]
  • Rotation Matrix in Space and its Determinant and EigenvaluesRotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
  • Dot Product, Lengths, and Distances of Complex VectorsDot Product, Lengths, and Distances of Complex Vectors For this problem, use the complex vectors \[ \mathbf{w}_1 = \begin{bmatrix} 1 + i \\ 1 - i \\ 0 \end{bmatrix} , \, \mathbf{w}_2 = \begin{bmatrix} -i \\ 0 \\ 2 - i \end{bmatrix} , \, \mathbf{w}_3 = \begin{bmatrix} 2+i \\ 1 - 3i \\ 2i \end{bmatrix} . \] Suppose $\mathbf{w}_4$ is […]
  • Construction of a Symmetric Matrix whose Inverse Matrix is ItselfConstruction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by \[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where […]
  • Unit Vectors and Idempotent MatricesUnit Vectors and Idempotent Matrices A square matrix $A$ is called idempotent if $A^2=A$. (a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$. Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$. Prove that $P$ is an idempotent matrix. (b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]
  • Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$ Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors. (a) Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$. (b) Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Linear algebra problems and solutions
True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces

Determine whether the following is true or false. If it is true, then give a proof. If it is false,...

Close