Vectors

[Cruger]

Hi Guys, I have a question like this.

1.
i.The straight line l passes through the points A and B with position vectors 7i - 3j + 6k and 10i + 3k respectively. The plane p has equation 3x - y +2z. Show that l is parallel to p.

ii. The point C is the foot of the perpendicular from A to p. Find a vector equation for the line which passes through C and is parallel to p.


I have answered the question i.

Line l has direction vector AB = (10i + 3k) - (7i - 3j + 6k) = 3i + 3j - 3k
Plane p has normal vector = 3i - j + 2k
(3i + 3j - 3k) • (3i - j + 2k) = 9 - 3 - 6 = 0
Since dot product of these two vectors = 9 - 3 - 6 = 0, then line is perpendicular to normal vector of plane.
Therefore, line l is parallel to the plane p

But i'm unaware of answering the question ii.
Plz help me to answer that.

Thank you.....

 

[pka]

Hi Guys, I have a question like this.

ii. The point C is the foot of the perpendicular from A to p. Find a vector equation for the line which passes through C and is parallel to p.


Since dot product of these two vectors = 9 - 3 - 6 = 0, then line is perpendicular to normal vector of plane. CORRECT
Therefore, line l is parallel to the plane p

But i'm unaware of answering the question ii.

Write the equation of the line thru \(\displaystyle A\) with the direction vector that of the normal to \(\displaystyle p\).
Find the point where that line intersects the plane.

 

[Cruger]

Write the equation of the line thru \(\displaystyle A\) with the direction vector that of the normal to \(\displaystyle p\).
Find the point where that line intersects the plane.

I didn't get that. can you do it?

 

[pka]

I didn't get that. can you do it?

No I will not do your work for you. That is your job.

What vector is normal to the plane \(\displaystyle p~?\).

If \(\displaystyle A\) is a point and \(\displaystyle \vec{N}\) is a vector, then what is the equation of the line thru \(\displaystyle A\) with direction \(\displaystyle \vec{N}~?\)

Do you know to find where a line insects a plane?

 

[DrPhil]

Hi Guys, I have a question like this.

1.
i.The straight line l passes through the points A and B with position vectors 7i - 3j + 6k and 10i + 3k respectively. The plane p has equation 3x - y +2z. Show that l is parallel to p.

ii. The point C is the foot of the perpendicular from A to p. Find a vector equation for the line which passes through C and is parallel to p.


I have answered the question i.

Line l has direction vector AB = (10i + 3k) - (7i - 3j + 6k) = 3i + 3j - 3k
Plane p has normal vector = 3i - j + 2k
(3i + 3j - 3k) • (3i - j + 2k) = 9 - 3 - 6 = 0
Since dot product of these two vectors = 9 - 3 - 6 = 0, then line is perpendicular to normal vector of plane.
Therefore, line l is parallel to the plane p

But i'm unaware of answering the question ii.
Plz help me to answer that.

Thank you.....

Write the equation of the line thru [FONT=MathJax_Math]A[/FONT] with the direction vector that of the normal to [FONT=MathJax_Math]p[/FONT].
Find the point where that line intersects the plane.

We do need to see your work to see where you are stuck.

You know point A, and you know the direction of the normal to the plane.
Let C = Xi + Yj + Zk. Since C is on the plane, 3X - Y + 2Z = 0
Direction vector AC = C - A must be a constant times the normal vector 3i - j + 2k [i.e., perpendicular to p]
You can express the line from A to C in parametric form as A + (direction)*t
Find C.

The statement "parallel to p" doesn't seem restrictive enough - are you sure the question is entered correctly? Since C is a point ON the plane, ANY line in the plane is parallel. Perhaps it should read "parallel to I"? If that were the case, the answer would be a line through point C with direction vector AB.