MY Vector Problems

[gta007]

My turn to post some vector problems which I have been unable to answer:

Q1) Find the point P on the line such that is parallel to the vector 3i + j.

Q2) Show that, if a vector in three dimensions make angles , and
respectively with the x, y and z axes then

[Mao]

Q1)


then the point P has the coordinates , and the vector OP will be

for it to be parallel to , there must be a constant when multiplied to OP, makes

or in other words, the vector OP must have the same ratio of i to j

hence,

and the rest is trivial

Q2

if you are using essentials book, theres a diagram on P71 that will help u:

for a vector

and etc...

that means:

[gta007]

Thank you very much

[gta007]

Okay more questions, I seem to have problems with the same types of question.

eg. A and B are defined by the position vectors a = 2i - 2j - k and b = 3i + 4k
Find the unit vector which bisects <AOB.

I already have

I found c, half way between a and b to be 2.5i - j + 1.5k. I don't know what to do from there.

The next question is more of the same.....
eg, A and B are points defined by the position vectors a = i + 3j - k and b = j + k
I have found the vector resolute of a in direction of b as j + k.

Then it tells me to find the unit vector through A perpendicular to OB. I have no idea.

Thirdly and lastly, I have trouble answering the questions which say find the shortest distance from x to line y etc.....

eg. A, B and C are points defined by the position vectors a = i + 2j + k, b = 2i + j - k and c = 2i - 3j + k

I found as i - j - 2k and as i - 5j
I then went on to find the vector resolute of in the line as
It then asks me to find the shortest distance from B to line AC. Here is where I am stuck.


Any help to help clarify these problems would be much appreciated.

[Mao]

1) <---omg cant believe i didnt realise it...
now that you have your two unit vectors, imagine them being the two sets of opposite sides of a rhombus. The angular bisector is easily the sum of the unit vectors, and then just find the unit vector of that

2)
vector resolutes are basically breaking vector a into rectangular components with b as an axis, which means:
a in the direction of b + a perpendicular to b = a

3)
this question follow the same logic as 2)
the shortest distance from B to AC is when the path is perpendicular to AC, or the vector of AB perpendicular to AC.

[Mao]

solutions:

1)
and

then the angular bisector can be found by arranging and like a rhombus, then finding the diagonals:





(why does that number look too complicated...?)


2)
let u be the vector resolute of parallel to :



let w be the vector of a perpendicular to






3)
let u be the vector resolute of in the direction of



let w be the vector of perpendicular to





shortest distance units

(why does this number also look too complicated...?)