10.1-10.5

(a) Find parametric equations for the line through (5, 3, 4) that is perpendicular to the plane x − y + 2z = 8. (Use the parameter t.) (b) In what points does this line intersect the coordinate planes?

(a) (x(t), y(t), z(t))=(5+t, 3-t, 4+2t) (b) xy-plane: (x, y, z) = (3,5,0) xz-plane: (x, y, z) =(0,8,-6) yz-plane: (x, y, z) = (8,0,10)

The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are |a| = 2 and |b| = 2. (a) Find |a × b|. (b) Use the right-hand rule to decide whether the components of a × b are positive, negative, or 0.

(a) 4 (b) x-component positive y-component negative z-component 0

Consider the points below. P(1, 0, 0), Q(0, 2, 0), R(0, 0, 3) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR.

(a) <5,3,2> (b) 7/2

State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (a) a · (b × c) (b) a × (b · c) (c) a × (b × c) (d) a · (b · c) (e) (a · b) × (c · d) (f) (a × b) · (c × d)

(a) The expression is meaningful. It is a scalar. (b) The expression is meaningless. The cross product is defined only for two vectors. (c) The expression is meaningful. It is a vector (d) The expression is meaningless. The dot product is defined only for two vectors. (e) The expression is meaningless. The cross product is defined only for two vectors. (f) The expression is meaningful. It is a scalar.

Determine whether each statement is true or false. (a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (d) Two planes perpendicular to a third plane are parallel. (e) Two lines parallel to a plane are parallel. (f) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two planes either intersect or are parallel. (j) Two lines either intersect or are parallel. (k) A plane and a line either intersect or are parallel.

(a) True (b) False (c) True (d) False (e) False (f) True (g) False (h) True (i) True (j) False (k) True

(a) What does the equation x = 3 represent in R²? What does it represent in R³? (b) What does the equation y = 2 represent in R³? What does z = 7 represent? What does the pair of equations y = 2, z = 7 represent? In other words, describe the set of points (x, y, z) such that y = 2 and z = 7.

(a) a line a plane (b) a plane a plane a line

Which of the following expressions are meaningful? Which are meaningless? Explain. (a) (a · b) · c (b) (a · b)c (c) |a|(b · c) (d) a · (b + c) (e) a · b + c (f) |a| · (b + c)

(a) has no meaning because it is the dot product of a scalar and a vector. (b) has meaning because it is the scalar multiple of a vector. (c) has meaning because it is the product of a scalar and a vector. (d) has meaning because it is the dot product of two vectors. (e) has no meaning because it is the sum of a scalar and a vector. (f) has no meaning because it is the dot product of a scalar and a vector.

Suppose that a ≠ 0. (a) If a · b = a · c, does it follow that b = c? (b) If a × b = a × c, does it follow that b = c? (c) If a · b = a · c, and a × b = a × c, does it follow that b = c?

(a) no (b) no (c) yes

Determine whether the given vectors are orthogonal, parallel, or neither. (a) u=<−6, 3, 6>, v=<8,−4,−8> (b) u=i-j+5k, v=5i-j+k (c) u=<a,b,c>, v=<-b,a,0>

(a) parallel (b) neither (c) orthogonal

Find the point at which the line x = 3 − t, y = 2 + t, z = 4t intersects the plane x − y + 3z = 11.

(x, y, z) = (2,3,4)

Find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c.

(x/a) + (y/b) + (z/c) =1

Find a · b. a= 4i+4j-k b= -5i+9k

-29

Find a · b. |a| = 2, |b| = 3, θ=2π/3 (the angle between a and b is 2π/3)

-3

Find the vector, not with determinants, but by using properties of cross products. (i × j) × k

0

Find an equation of the plane. The plane through the origin and the points (2, −3, 8) and (7, 3, 2)

0=-30x+52y+27z

Find an equation of the sphere with center (3, −8, 4) and radius 5. Use an equation to describe its intersection with each of the coordinate planes. (If the sphere does not intersect with the plane, enter DNE.)

25=(x-3)²+(y+8)²+(z-4)² Intersection with xy-plane: 9=(x-3)²+(y+8)² Intersection with xz-plane: DNE Intersection with yz-plane: 16=(y+8)²+(z-4)²

Find a · b. a= <4,1,1/5> b= <9,-2,-15>

31

Find an equation of the sphere that passes through the point (4, 1, −3) and has center (3, 8, 1)

66=(x-3)²+(y-8)²+(z-1)²

Find the sum of the given vectors. a= <3,-2>, b= <-2,6> Illustrate geometrically.

<1,4>

Find a unit vector that is orthogonal to both i + j and i + k.

<1/√3,-1/√3,-1/√3>

Find the cross product a × b. a=i+3j-3k b=-i+4k Verify that it is orthogonal to both a and b.

<12,-1,3> (a × b)·a=0 (a × b)·b=0

Find a vector a with representation given by the directed line segment AB. A(0, 1, 2), B(3, 1, −3) Draw AB and the equivalent representation starting at the origin.

<3,0,-5>

Find the cross product a × b. a= <9,0,-4> b= <0,9,0> Verify that it is orthogonal to both a and b.

<36,0,81> (a × b)·a=0 (a × b)·b=0

Find a vector a with representation given by the directed line segment AB. A(−5, 3), B(1, 4) Draw AB and the equivalent representation starting at the origin.

<6,1,0>

Find a + b, 2a + 3b, |a|, and |a − b|. a = 2i − 5j + 4k, b = 2j − k

a+b= <2,-3,3> 2a+3b=<4,-4,5> |a|=√45 |a-b|=√78

Find a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (0, 11, −6) and parallel to the line x = −1 + 2t, y = 6 − 3t, z = 3 + 8t

r(t)=<0,11,-6> + t<2,-3,8> (x(t), y(t), z(t))=(2t, -11-3t, -6+8t)

Find a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (4, −9, 6) and parallel to the vector <1,5,−2/3>

r(t)=<4,-9,6> + t<1,5,-2/3> (x(t), y(t), z(t))=(4+t, -9+5t, 6-(2t/3))

If v lies in the first quadrant and makes an angle π/3 with the positive x-axis and |v| = 4, find v in component form.

v= <2,2√3,0>

Find an equation of the plane. The plane through the points (0, 7, 7), (7, 0, 7), and (7, 7, 0)

x+y+z=14