Solution: In parts (a)(g) determine whether the statement is true or false, and justify

In Exercises 12, determine whether u and v are orthogonal vectors.

In Exercises 12, determine whether u and v are orthogonal vectors.

In Exercises 34, determine whether the vectors form an orthogonal set.

Find a unit vector that is orthogonal to both and

(a) Show that and are orthogonal vectors. (b) Use the result in part (a) to find two vectors that are orthogonal to . (c) Find two unit vectors that are orthogonal to .

Do the points , , and form the vertices of a right triangle? Explain your answer.

Repeat Exercise 7 for the points , , and .

In Exercises 912, find a point-normal form of the equation of the plane passing through P and having n as a normal.

In Exercises 912, find a point-normal form of the equation of the plane passing through P and having n as a normal.

In Exercises 912, find a point-normal form of the equation of the plane passing through P and having n as a normal.

In Exercises 912, find a point-normal form of the equation of the plane passing through P and having n as a normal.

In Exercises 912, find a point-normal form of the equation of the plane passing through P and having n as a normal.

In Exercises 1316, determine whether the given planes are parallel.

In Exercises 1316, determine whether the given planes are parallel.

In Exercises 1316, determine whether the given planes are parallel.

In Exercises 1316, determine whether the given planes are parallel.

In Exercises 1718, determine whether the given planes are perpendicular.

In Exercises 1718, determine whether the given planes are perpendicular.

In Exercises 1920, find .

In Exercises 1920, find .

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2128, find the vector component of u along a and the vector component of u orthogonal to a.

In Exercises 2932, find the distance between the point and the line.

In Exercises 2932, find the distance between the point and the line.

In Exercises 2932, find the distance between the point and the line.

In Exercises 2932, find the distance between the point and the line.

In Exercises 3336, find the distance between the point and the plane.

In Exercises 3336, find the distance between the point and the plane.

In Exercises 3336, find the distance between the point and the plane.

In Exercises 3336, find the distance between the point and the plane.

In Exercises 3740, find the distance between the given parallel planes.

In Exercises 3740, find the distance between the given parallel planes.

In Exercises 3740, find the distance between the given parallel planes.

In Exercises 3740, find the distance between the given parallel planes.

Let i, j, and k be unit vectors along the positive x, y, and z axes of a rectangular coordinate system in 3-space. If is a nonzero vector, then the angles , , and between v and the vectors i, j, and k, respectively, are called the direction angles of v (Figure Ex-41), and the numbers , and are called the direction cosines of v. (a) Show that . (b) Find and . (c) Show that . (d) Show that . Figure Ex-41

Use the result in Exercise 41 to estimate, to the nearest degree, the angles that a diagonal of a box with dimensions makes with the edges of the box

Show that if v is orthogonal to both and , then v is orthogonal to for all scalars and .

Let u and v be nonzero vectors in 2- or 3-space, and let and . Show that the vector bisects the angle between u and v

Prove part (a) of Theorem 3.3.4.

Is it possible to have

In parts (a)(g) determine whether the statement is true or false, and justify your answer.The vectors and are orthogonal.

In parts (a)(g) determine whether the statement is true or false, and justify your answer.If u and v are orthogonal vectors, then for all nonzero scalars k and m, and are orthogonal vectors.

In parts (a)(g) determine whether the statement is true or false, and justify your answer.The orthogonal projection of u along a is perpendicular to the vector component of u orthogonal to a.

In parts (a)(g) determine whether the statement is true or false, and justify your answer. If a and b are orthogonal vectors, then for every nonzero vector u, we have

In parts (a)(g) determine whether the statement is true or false, and justify your answer.If a and u are nonzero vectors, then

In parts (a)(g) determine whether the statement is true or false, and justify your answer. If the relationship holds for some nonzero vector a, then .

In parts (a)(g) determine whether the statement is true or false, and justify your answer.For all vectors u and v, it is true that