Study guide and
13 practice problems
on:
Dot product of perpendicular vectors is zero
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The vectors $\mathbf{x}$ and $\mathbf{y}$ are perpendicular if and only if $\mathbf{x} \cdot \mathbf{y} = 0$.
Related topics
Dot product
(41 problems)
Multivariable calculus
(147 problems)
Practice problems
Find a 2d vector that is perpendicular to $\langle 2,3 \rangle$. Verify that it is perpendicular.
Solution
If $\bfx \cdot \bfy = \bfx \cdot \bfz$ for a nonzero $\bfx$, does $\bfy = \bfz$? If so, prove it. If not, provide a counterexample.
Solution
Of the unit vectors $\mathbf{A}$ through $\mathbf{H}$,
Which have a positive dot product with $\mathbf{A}$?
Which have a negative dot product with $\mathbf{A}$?
Which have zero dot product with $\mathbf{A}$?
Which has the largest dot product with $\mathbf{A}$?
Which has the most negative dot product with $\mathbf{A}$?
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
Consider the point $\bfx_0 = (x_0, y_0)$ and the line given by $\bfn \cdot \bfx = 0$, where $\bfn = (a, b)$. Using a vector projection, find the coordinates of the nearest point to $\bfx_0$ on the line $\bfn\cdot \bfx =0$.
Solution
Consider the point $\bfx_0 = (x_0, y_0)$ and the line given by $\bfn \cdot \bfx = 0$, where $\bfn = (a, b)$. Show that the minimum distance from $\bfx_0$ to the line $\bfn\cdot\bfx=0$ is $\frac{\left |\bfn\cdot \bfx_0 \right|}{\left| \bfn \right| }$.
Solution
Show that $\bfa \times \bfb$ is perpendicular to $\bfa$ by computing a dot product.
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
Consider the surface $x^3 + y^3 z = 3$. Find tangent vector at the point $(1,1,2)$ that has $\mathbf{i}$ component 1 and $\mathbf{j}$ component 1. To find it, first find a normal vector.
Solution
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