91影视

Let $\mathbf{v}=a\mathbf{i}+b\mathbf{j}$and $\mathbf{w}=c\mathbf{i}+d\mathbf{j}$. Give conditions on the constants $a,b,c$, and $d$that guarantee that

(a) $\mathbf{v}$is parallel to $\mathit{w}$.

(b) $\mathbf{v}$is perpendicular to $\mathbf{w}$.

Explain why there are infinitely many different planes containing any given line in ${\mathrm{鈩�}}^3$What form does the equation of a plane containing the $x-$axis have?

Find the equation of the sphere center at \((2,-3,4)\) and tangent to the \(xz\) plane.

The sides of a 2 脳 3 脳 4 rectangular solid are parallel to the coordinate planes. The coordinates of four of its vertices are (1, 鈭�2, 3), (鈭�1, 鈭�2, 鈭�1), (鈭�1, 1, 3), and (1, 鈭�2, 3). What are the coordinates of the other four vertices?

In Exercises 3鈥�8, let A, B, C, D, ... , Z points in $R^3$. Simplify the given quantity.

role="math" localid="1663305734277" $\stackrel{鈫�}{BD}-\stackrel{鈫�}{CD}$

What is Lagrange鈥檚 identity? How is it used to understand the geometry of the cross product?

Turning a bolt with a wrench produces a torque vector that drives the bolt forward. The magnitude of the torque vector is $||r||||\mathbf{F}||\sin 胃$, where r is the vector along the handle of the wrench, F is the force vector applied to the handle of the wrench, and 胃 is the angle between these two vectors. Therefore, the magnitude of the torque is $||r脳\mathbf{F}||$. In Exercises 60 and 61, find the magnitude of the torque. Express each answer in foot-pounds.

A force of 20 lb is applied to a wrench with a 6-inch handle at an angle of ${60}^{鈼�}$.

Let \(胃\) be the angle between nonzero vectors \(u\) and \(v\).

Prove each of the following:

(a) \(胃\) is acute if and only if \(u 路 v > 0\)

(b) \(胃\) is right if and only if \(u 路 v = 0\)

(c) \(胃\) is obtuse if and only if \(u 路 v < 0\)

Prove Theorem 10.11; that is, show that when $v鈮�0$, the scaled vector role="math" localid="1663644167280" $\frac{1}{\left|v\right|}v$ is a unit vector with the same direction as $v$.

The Subaru reflecting telescope on Mauna Kea, Hawaii, has a mirror in the shape of a circular paraboloid with a diameter of 8.3 meters and a focal length of 15 meters. While there are some tricks to how that focal length plays out in practice, if we put the center of the telescope at the origin, pointed straight up, then the effective focus would be at (0, 0, p), where p satisfies \(4pz=x^{2}+y^{2}\), with all distances given in meters. How high above the xy-plane is the edge of the telescope?