Math 280 Calculus III Chapter 12 Vectors and the Geometry of Space

Sections: 12.1, 12.2, 12.3 12.4, 12.6 12.5 Topics:

Three-Dimensional Coordinate Systems

Vectors

The Dot Product

The Cross Product

Cylinders and Quadric Surfaces

Lines and Planes in Space

Section 12.1 Three-Dimensional Coordinate Systems

Example

For figure 12.4 shown above, find the distance between the two points (2, 0, 0) and (0, 2, 3).

__________ Complete the square to verify that the results here are correct.

Example

Describe the sets of points in space whose coordinates satisfy the given inequalities or

combinations of equations and inequalities.

(a) �� � �� � 1�� � 0

(b) �� � �� � 1�� � 3

(c) �� � �� � 1��������������

Example

Describe the given set with a single equation or pair of equations.

The set of points equidistant from the origin and the point (0, 2, 0).

Example

Find an equation for the set of all points equidistant from the point (0, 0, 2) and the

xy-plane.

Section 12.2 Vectors

Example Find the component form and length of the vector from I(−3, 4, 1) to T(−5, 2, 2).

Example A small cart is being pulled along a smooth horizontal floor with a 20-lb force F

Making a 45 degree angle to the floor (figure 12.11). What is the effective force

moving the cart forward?

Solution: The effective force is the horizontal component of F = ⟨�, �⟩, given be

� � |�| cos 45 � "20$ %√22 ' ( 14.14*�.

The linear algebraic notion of a vector that you have probably studied in previous courses

becomes relevant in calculus in applications such as the tangent of the velocity vector

defined at a point on the velocity vector curve.

We can compare the concepts in figure 12.8 to the concepts in figure 12.11 above.

Example 4 Find a unit vector in the direction of the vector from I(1, 0, 1) to T(3, 2, 0).

Example 5 If v = 3i – 4j is a velocity vector, express v as a product of its speed times a unit

vector in the direction of motion.

Example 6 A force of 6 newtons is applied in the direction of the vector v = 2i +2j – k.

Express the force F as a product of its magnitude and direction.

Example 8 A jet airliner, flying due east at 500 mph in still air, encounters a 70-mph

tailwind blowing in the direction 60 degrees north of east. The airplane holds

its compass heading due east but, because of the wind, acquires a new ground

speed and direction. What are they?

Example 9 A 75-N weight is suspended by two wires, as shown in Figure 12.18a. Find the

Forces F1 and F2 acting in both wires.

Section 12.3 The Dot Product

Note that the solution of a dot product of two vectors is a scalar value.

Example 3 Find the angle θ in the triangle ABC determined by the vertices

A = (0, 0), B = (3, 5), and C = (5, 2). (figure 12.22 above)

Example 4 Determine whether the vectors are orthogonal.

(a) , � ⟨3, −2⟩, - � ⟨4, 6⟩

(b) , � 3/ − 20, - � 20 � 41

(c) Any vector u and the vector 0.

Example 5 Find the vector projection of, � 6/ � 30 � 21���- � 2 − 20 − 21 and the scalar component of u in the direction of v.

Example 7 If |3| = 40 N (newtons) , |4| = 3 m , and θ = 60°, find the work done by F

acting from P to Q.

Example

Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of

the vector v is the negative reciprocal of the slope of the given line.

Section 12.4 The Cross Product

Example

Find the length and direction of u × v. , � 7� 2 − 8

� 9 � :,- � 2 � 9 � 2:

Example 1 Find u × v and v × u using the determinant. , � 22 � 9 � :, - � −42 � 39 � :

Example 2

Find a vector perpendicular to the plane of P(1, −1, 0), Q(2, 1, −1), and R(−1, 1, 2).

Example 3

Find the area of the triangle with vertices P(1, −1, 0), Q(2, 1, −1), and R(−1, 1, 2).

Example 4

Find a unit vector perpendicular to the plane of P(1, −1, 0), Q(2, 1, −1), and R(−1, 1, 2).

Section 12.6 Cylinders and Quadric Surfaces (section 12.5 is presented later)

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A cylinder is a surface that is generated by moving a straight line along a given planar

curve while holding the line parallel to a given fixed line.

.

Example Sketch the surface defined by the equation z = y2 – 1.

Example Sketch the surface defined by the equation z = x2 + 4y2.

Example Sketch the surface defined by the equation 1 = x2 + y2 – z2 .

Section 12.5 Lines and Planes in Space

Example 4 A helicopter is to fly directly from a helipad at the origin in the direction of the

point (1, 1, 1) a the speed of 60 ft/sec. What is the position of the helicopter after 10 sec?

Example 6

Find the equation for the plane through Po(−3, 0, 7) perpendicular to n = 5i + 2j – k.

Example 7 Find an equation for the plane through A(0, 0, 1), B(2, 0, 0), and C(0, 3, 0).

Example 8 Find a vector parallel to the line of intersection of the planes

3x – 6y – 2z = 15 and 2x + y – 2z = 5.

Example 9 Find parametric equations for the line in which the planes 3x – 6y – 2z = 15

and 2x + y – 2z = 5. intersect.

Example 10 Find the point where the line � � ;7� 2�,� = −2�,� = 1 + �

intersects the plane 3x + 2y + 6z = 6.

Example 11 Find the distance from S(1, 1, 3) to the plane 3x + 2y + 6z = 6.

Example 12 Find the angle between the planes 3x – 6y – 2z = 15 and 2x + y – 2z = 5.