Study guide and
21 practice problems
on:
Direct computation of dot product
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If $\mathbf{x} = \langle x_1, x_2, x_3 \rangle$ and $\mathbf{y} = \langle y_1, y_2, y_3 \rangle$, the dot product is defined as $$ \mathbf{x} \cdot \mathbf{y} = x_1 y_1 + x_2 y_2 + x_3 y_3.$$
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
Prove that $\cos (\theta_1 + \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2$ by considering the dot product of the unit vectors $\mathbf{v_1}$ and $\mathbf{v_2}$. These vectors are at the angles $\theta_1$ above the $x$-axis and $\theta_2$ below the $x$-axis, respectively.
Solution
Find the angle at the apex of a triangular faces of the pyramid formed by the points $(1, 1, 0)$, $(1,-1, 0)$, $(-1, 1, 0)$, $(-1, -1, 0)$, and $(0,0,1)$.
Solution
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
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
Prove for 2d vectors: $\bfa \cdot (\bfb + \bfc) = \bfa\cdot \bfb + \bfa \cdot \bfc.$
Solution
Compute the vector projection of $\bfi$ onto $\bfi + \bfj$.
Solution
A block rests on an inclined plane of angle $\theta$, as shown. Gravity provides a force $\bfF = (0, -m g)$ on the block.
(a) What is the component of $\bfF$ in the downhill direction?
(b) What is the projection of $\bfF$ in the same direction?
Solution
Sketch and find the volume of the parallelepiped spanned by the vectors $\bfi, \bfi+\bfj,$ and $\bfi + \bfk.$
Solution
Show that $\bfa \times \bfb$ is perpendicular to $\bfa$ by computing a dot product.
Solution
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