Stewart Precalc 6e Section 9.ConceptCheck

What is the projection of \(\mathbf{u}\) onto \(\mathbf{v}\), and how do you calculate it?

How much work is done by the force \(\mathbf{F}\) in moving an object along a displacement \(\mathbf{D}\)?

How do you find the distance between two points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) in three-dimensional space?

What is the equation of the sphere with center \(C(a, b, c)\) and radius \(r\)?

(a) How do you calculate the cross product \(\mathbf{a} \times \mathbf{b}\) if you know the components of \(\mathbf{a}\) and \(\mathbf{b}\)? (b) How do you calculate \(\mathbf{a} \times \mathbf{b}\) if you know the lengths of \(\mathbf{a}\) and \(\mathbf{b}\) and the angle between them? (c) What is the angle between \(\mathbf{a} \times \mathbf{b}\) and each of \(\mathbf{a}\) and \(\mathbf{b}\)?

(a) How do you find the area of the parallelogram determined by \(\mathbf{a}\) and \(\mathbf{b}\)? (b) How do you find the volume of the parallelepiped determined by \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\)?

Write parametric equations for the line that contains the point \(P(x_0, y_0, z_0)\) and is parallel to the vector \(\mathbf{v} = (a, b, c)\).

Write an equation for the plane that contains the point \(P(x_0, y_0, z_0)\) and has normal vector \(\mathbf{n} = (a, b, c)\).

How do you find parametric equations for the line that contains the points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\)?

How do you find an equation for the plane that contains the points \(P(x_1, y_1, z_1)\), \(Q(x_2, y_2, z_2)\), and \(R(x_3, y_3, z_3)\)?