# Vectorial Algebra: Parallel, Perpendicular or Neither?

• sdoyle
In summary, the line passing through the point P=(1,-1,1) with direction vector d=[2,3,-1] is neither parallel nor perpendicular to the plane 2x+3y-z=1. The point P and directional vector d are independent and cannot be related. The normal vector of the plane is needed to determine if the line and the plane are parallel or perpendicular. These types of questions may be better suited for a math homework forum.
sdoyle

## Homework Statement

The line l passes through the point P=(1,-1,1) and has direction vector d=[2,3,-1]. Determine whether l and P are parallel, perpendicular, or neither to 2x+3y-z=1.

## Homework Equations

n.x=n.p, also cross product for parallel lines, and dot product for perpendicular.

## The Attempt at a Solution

I don't know how to relate the point, P, to the directional vector,d. I am pretty sure that I can do the rest.

P isn't related to d, they are independent. 2x+3y-z=1 is a plane. What it's normal vector? Question like this probably don't belong in Physics. I'd put them in a math HW forum.

I would approach this problem by first understanding the concept of vectorial algebra and how it relates to parallel and perpendicular lines. In this case, the direction vector d represents the slope of the line l, while the equation 2x+3y-z=1 represents a plane in 3-dimensional space.

To determine whether l and P are parallel or perpendicular to the plane, we can use the dot product and cross product. The dot product of the normal vector of the plane (2,3,-1) and the direction vector d (2,3,-1) should be equal to zero for the line to be perpendicular to the plane. Similarly, the cross product of the normal vector and the direction vector should be equal to zero for the line to be parallel to the plane.

Next, we can use the point P and the equation of the plane to determine if P is on the plane. If P satisfies the equation, then it lies on the plane and is therefore parallel to the line l.

In conclusion, by using vectorial algebra and the dot and cross products, we can determine whether the line l and point P are parallel, perpendicular, or neither to the given plane.

## 1. What is a vector in algebra?

A vector in algebra is a mathematical object that has both magnitude and direction. It is typically represented as an arrow pointing in a specific direction, with the length of the arrow representing the magnitude or size of the vector.

## 2. What does it mean for two vectors to be parallel?

Two vectors are parallel if they have the same direction or are in the same line. This means that they will never intersect, and their direction will remain the same no matter how far they are extended.

## 3. How can I determine if two vectors are perpendicular?

Two vectors are perpendicular if their dot product is equal to zero. This means that the angle between the two vectors is 90 degrees, forming a right angle.

## 4. Can two vectors be both parallel and perpendicular?

No, two vectors cannot be both parallel and perpendicular. If two vectors are parallel, their dot product will never be equal to zero, making them not perpendicular. Similarly, if two vectors are perpendicular, they will never have the same direction, making them not parallel.

## 5. How can vector algebra be useful in real-life applications?

Vector algebra is used in many real-life applications, such as physics, engineering, and navigation. It allows us to represent and manipulate quantities that have both magnitude and direction, making it useful in solving problems involving motion, forces, and geometric relationships.

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