Solve Vector Problems: Scalar Product, Perpendicularity & Distance

[vorcil]

I need help in knowing how to solve these, i'd put down how in the next post

1:scalar product of two vectors: (1,2,-2) and (1,-2,2)

2. One of: “If non-zero vectors x and y are perpendicular, then”
“If non-zero vectors x and y are parallel, then”
“If x is a unit vector with same direction and sense as y, then”
a) x = y, b) x = ky for some scalar k, c) x = y / |y|, d) x . y = 0, e) x . y = 1

3. Find cosθ, where θ is the angle between a = (1, 2, 4) and b = (4, –2, 1).

4. Write down a parametric equation of a given straight line.

5. A direction perpendicular to the plane 2x – y + z = 9 is:
a) (2x, –y, z), b) (4, 0, 1), c) (1, 1, –1), d) (2, –1, 1).

6. If the plane 5x + y – 3z = k contains the point (1, 4, 2) then k = ?

7. The line x = (–1, 2, 4) + t(5, 1, 0) meets the plane y = 0 at the point (?, 0, ?).

8. The projection of the vector (4, 0, 7) onto the direction (–1, –2, 2) is ?

9. The distance from the point (1, 2, –5) to the plane 2x + y – 2z = 8 is ?

10. Find the speed of a particle at a given time, given the position vector as a function of time.

 

[vorcil]

1: 1,2,-2 * 1,-2,2
1 -4 -4 = -7? is that how you find the scalar product of those two vectors?

how do i do number 2:??

3: find cos angle between the two vectors
2,3,2 & 2,-2,3
a & b
a.b = 4
squareroot 17 * square root of 17

so it's cos theta = 4/17

 

[vorcil]

4: direction vector (1,1,-2) point on line (1,-2,2)

the equation would be, X=(1,-2,2) + t(1,1,-2)?
which is the same as
(1+t, -2+t, 2-2t)?

 

[vorcil]

5. A direction perpendicular to the plane 2x – y + z = 9 is:
a) (2x, –y, z), b) (4, 0, 1), c) (1, 1, –1), d) (2, –1, 1).

 

[belliott4488]

1: 1,2,-2 * 1,-2,2
1 -4 -4 = -7? is that how you find the scalar product of those two vectors?

Looks good to me!

how do i do number 2:??

I don't understand the problem statement. There are three "if" conditions, and then a bunch of choices for possible conclusions. I would say that each of those three "if" conditions matches one of the conclusions a), b), c), d), or e). Is that what you're supposed to answer? Three choices?

3: find cos angle between the two vectors
2,3,2 & 2,-2,3
a & b
a.b = 4
squareroot 17 * square root of 17

so it's cos theta = 4/17

Yup!

 

[belliott4488]

4: direction vector (1,1,-2) point on line (1,-2,2)

the equation would be, X=(1,-2,2) + t(1,1,-2)?
which is the same as
(1+t, -2+t, 2-2t)?

I'll buy that.

 

[vorcil]

Looks good to me!

I don't understand the problem statement. There are three "if" conditions, and then a bunch of choices for possible conclusions. I would say that each of those three "if" conditions matches one of the conclusions a), b), c), d), or e). Is that what you're supposed to answer? Three choices?
Yup!

The things i gave above are what the questions I'm going to be given are based upon,
So i'll be given a question about

One of:
“If non-zero vectors x and y are perpendicular, then”
“If non-zero vectors x and y are parallel, then”
“If x is a unit vector with same direction and sense as y, then”

a) x = y,
b) x = ky for some scalar k,
c) x = y / |y|,
d) x . y = 0,
e) x . y = 1

so if it's the first one
“If non-zero vectors x and y are perpendicular, then”
what would i put for those awnsers a,b,c,d,e

“If non-zero vectors x and y are parallel, then”
what'd i put?

“If x is a unit vector with same direction and sense as y, then”

 

[vorcil]

The projection of the vector (4, 0, 7) onto the direction (–1, –2, 2) is

Directions' D^ = (1/3)*(-1,-2,2)
so 4*-1, 0*-2, 2*7) scalar product of those vectors divided by 3?

awnser is 10/3?

 

[vorcil]

The things i gave above are what the questions I'm going to be given are based upon,
So i'll be given a question about

One of:
“If non-zero vectors x and y are perpendicular, then”
“If non-zero vectors x and y are parallel, then”
“If x is a unit vector with same direction and sense as y, then”

a) x = y,
b) x = ky for some scalar k,
c) x = y / |y|,
d) x . y = 0,
e) x . y = 1

so if it's the first one
“If non-zero vectors x and y are perpendicular, then”
what would i put for those awnsers a,b,c,d,e

“If non-zero vectors x and y are parallel, then”
what'd i put?

“If x is a unit vector with same direction and sense as y, then”

FIGURED it out

if it's perpendicular
x.y = 0

if it's parallel
x = Ky

for some unit vector with the same direction and sense as y
x^ = y / | y |
since they are the same, x^ = y^ and x^ = x/ | x |