Study guide and
7 practice problems
on:
Dot product and vector length
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The dot product of a vector with itself gives the squared length of that vector:$$\mathbf{x} \cdot \mathbf{x} = |\mathbf{x}|^2.$$
Related topics
Length of a vector
(32 problems)
Dot product
(41 problems)
Multivariable calculus
(147 problems)
Practice problems
Prove that $\bfx \cdot \bfx = \left| \bfx \right|^2$ in two ways:
Directly (in the case of 3d vectors)
By the dot product angle formula
Solution
Derive the law of cosines using the dot product:
(a) Write $\text{CB}$ in terms of $\text{OB}$ and $\text{OC}$
(b) Write $\left | \text{CB} \right|^2$ in terms of $\left | \text{OB} \right |$, $\left | \text{OC} \right |$ and $\text{OB} \cdot \text{OC}$
(c) Show that $\left | \text{CB} \right|^2 = \left | \text{OB} \right|^2 + \left | \text{OC} \right |^2 - 2 \left | \text{OB} \right| \left | \text{OC} \right| \cos \theta$.
Solution
Show that the lines connecting any point on the semicircle of radius 1 to $(1,0)$ and $(-1,0)$ are perpendicular.
Solution
Use vectors and dot products to prove: if the diagonals of a rectangle are perpendicular, then the rectangle is a square.
Solution
Prove the parallelogram law: The sum of the squares of the lengths of both diagonals of a parallelogram equals the sum of the squares of the lengths of all four sides.
Solution
An orthogonal matrix is one satisfying $A A^t = I$. Suppose $$A = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ a & b & c \end{pmatrix}.$$
If $A$ is orthogonal, show that $(a, b, c)$ is perpendicular to $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0)$ and $(0,\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$
If $A$ is orthogonal, show that $(a,b,c)$ is of unit length.
Find two values of $(a, b, c)$ so that $A$ is orthogonal.
Solution
An ant is on a merry-go-ground that is rotating clockwise at $\omega$ radians per second. Initially, the ant is at $(R,0)$. From the ant's perspective, it walks toward the center with speed $v$. Several snapshots in time are as follows:
Find the parameterization of the path taken by the ant (relative to the ground)
Compute the speed of the ant as a function of $t$. When is it largest?
Set up, but do not evaluate, an integral for the arc length of the path taken by the ant between $t=0$ and when the ant reaches the origin
Solution