# Angle between a plane and a line

• Krushnaraj Pandya
In summary, the problem involves finding the value of 3a+3b given that the plane 3x+y+2z+6=0 is parallel to the line (3x-1)/2b = 3-y = (z-1)/a. Using the formula for the angle between two lines, it can be determined that 2b/3 and a are the direction ratios for the line and 3,1,2 are the direction ratios for the plane. By setting these two vectors perpendicular to each other, the equation b+3a=1 can be derived. However, this is not enough information to solve for a and b separately.
Krushnaraj Pandya
Gold Member

## Homework Statement

If the plane 3x+y+2z+6=0 is parallel to the line (3x-1)/2b = 3-y = (z-1)/a then 3a+3b is?

## Homework Equations

angle between two lines=90 degree when l1l2+m1m2+n1n2=0 where l,m,n are direction ratios...(i)

## The Attempt at a Solution

First I divided x/3 and 3-y by -1 to get the direction ratios 2b/3; -1 and a. Since the plane is parallel the normal to the plane is perpendicular to the line. Using 3,1,2 for the plane's dr's and putting it in (i) we get b+3a=1, I'm stuck here. I tried to use AM>=GM but that gave me ab<=1/12 which isn't very useful. How do I get a second equation?

Krushnaraj Pandya said:

## Homework Statement

If the plane 3x+y+2z+6=0 is parallel to the line (3x-1)/2b = 3-y = (z-1)/a then 3a+3b is?

## Homework Equations

angle between two lines=90 degree when l1l2+m1m2+n1n2=0 where l,m,n are direction ratios...(i)

## The Attempt at a Solution

First I divided x/3 and 3-y by -1 to get the direction ratios 2b/3; -1 and a. Since the plane is parallel the normal to the plane is perpendicular to the line. Using 3,1,2 for the plane's dr's ? and putting it in (i) we get b+3a=1, I'm stuck here. I tried to use AM>=GM but that gave me ab<=1/12 which isn't very useful. How do I get a second equation?
What is the normal of the plane? What is the directional vector of the line? And they are perpendicular...
I think you have some mistake when calculating the scalar product.

Krushnaraj Pandya said:

## Homework Statement

If the plane 3x+y+2z+6=0 is parallel to the line (3x-1)/2b = 3-y = (z-1)/a then 3a+3b is?

## Homework Equations

angle between two lines=90 degree when l1l2+m1m2+n1n2=0 where l,m,n are direction ratios...(i)

## The Attempt at a Solution

First I divided x/3 and 3-y by -1 to get the direction ratios 2b/3; -1 and a. Since the plane is parallel the normal to the plane is perpendicular to the line. Using 3,1,2 for the plane's dr's and putting it in (i) we get b+3a=1, I'm stuck here. I tried to use AM>=GM but that gave me ab<=1/12 which isn't very useful. How do I get a second equation?
You need ##(2b/3,-1,a) \perp (3,1,2).## How do you express that algebraically? What condition do you get on ##a## and ##b##?

Note: if you think about the problem geometrically you will see that you were not given enough information to determine ##a## and ##b## separately, but can at least get a relationship between them.

## 1. What is the angle between a plane and a line?

The angle between a plane and a line is the measure of the deviation from perpendicularity between the two objects. It is the smallest angle formed between the line and a line drawn on the plane that is perpendicular to the line.

## 2. How is the angle between a plane and a line calculated?

The angle between a plane and a line can be calculated using the dot product between the normal vector of the plane and the direction vector of the line. The formula is given as: θ = cos⁻¹(|a · b| / |a||b|), where a is the normal vector of the plane and b is the direction vector of the line.

## 3. Can the angle between a plane and a line be negative?

No, the angle between a plane and a line is always positive. It represents the smallest angle between the two objects, so it cannot be negative.

## 4. What does a small angle between a plane and a line indicate?

A small angle between a plane and a line indicates that the two objects are close to being perpendicular, meaning they are nearly at a right angle to each other. This can be useful in determining the orientation and alignment of objects in three-dimensional space.

## 5. How does the angle between a plane and a line affect the intersection between the two objects?

The angle between a plane and a line does not directly affect the intersection between the two objects. However, if the angle is 0 degrees (or very close to 0), it means the line lies on the plane and there is an infinite number of intersection points. If the angle is 90 degrees (or very close to 90), it means the line is parallel to the plane and there is no intersection point.

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