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Chapter 10. Vectors and the Geometry of Space
§ 10.3. The Dot Product
- Find ∙ .
(a)
(b)
(c) , the angle between and is
- Find the scalar and vector projection of onto .
(a) ,
(b) ^
- Show that the vector or
is orthogonal to .
4. Use a scalar projection to show that the distance from a point
(^) to the line
is
- If ,
and
, show that the vector equation
∙ represents a sphere and find its center and radius.
- Show that if and are orthogonal, then the vectors and must have the same length.
§ 10.4. The Cross Product
- Find two unit vectors orthogonal to both 〈 〉 and 〈 〉.
2. (a) Find the area of the parallelogram with vertices , , , and .
(b) Find a non-zero vector orthogonal to the plane through the points , ,
, and find the area of triangle .
3. Find the volume of the parallelepiped with adjacent edges , , ;
- Use the scalar triple product to verify that the vectors
, and
are coplanar.
- (a) Find all vectors
such that 〈 〉×
(b) Explain why there is no vector
such that 〈 〉×
6. (a) Let be a point not on the line that passes through points and . Show that the
distance from the point to the line is
×
^
where
and
(b) Use the formula in part (a) to find the distance from the point to the line through
and .
7. (a) Let be a point not on the plane that passes through the points , , and . Show that the
distance from the point to the plane is ^
×
×
^
where
^
^
^
(b) Use the formula in part (a) to find the distance from the point ^ to the plane through the
points , ^ and .
- Suppose that
(a) If
, does it follow that
(b) If
×
×
, does it follow that
(c) If
and
×
×
, does it follow that
- Determine whether the statement is true or false. If it is true, give a brief reason. If it is false,
give a counter example or disprove it.
(a) For any vectors
^
×
^
^
^
^
(b) For any vectors
^
×
(c) For any vectors
, ^
≤ ^
^
- (a) Prove that
×
×
(b) Use (a) to prove that
×
×
×
×
×
×
(c) Prove that ^
×
⋅ ^
×
^
^
^
^
§ 10.5. Equations of lines and planes
- Find the distance between the skew lines with parametric equations , , ,
and , , .
2.(a) Find an equation of the plane that passes through the points , and
(b) Find symmetric equations for the line through that is perpendicular to the plane in part (a).
(c)A second plane passes through and has normal vector . Find the acute
angle of intersection of the planes to the nearest degree.
