# 3d vectors. Does a point lie on the line?

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In summary, a question is posed about a vector equation and whether a given point lies on the line. The person asks for help and suggests rewriting the line in a different form. It is then suggested to write the line in parametric form.

## Homework Statement

L1 has the vector equation r=2i+3j-5k + t(4i-2j+k)
Does the point (16,-3,-2) lie on the line?

Unknown

## The Attempt at a Solution

I have a simple(i think) vector question but i just don't know the method to work it out.

I have searched on the internet and through my books and i can't find a similar problem anywhere. Any help would be much appreciated.

Thanks

The point (16, -3, -2) can also be written as 16i - 3j - 2k. Does that help? Try writing the line in the similar form r = Ai + Bj + Ck where A, B, and C may be functions of t.

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Conversely, the vector equation for the line r=2i+3j-5k + t(4i-2j+k) is the same as the parametric equation x= 2+ 4t, y= 3- 2t, z= -5+ k.

## 1. What are 3D vectors?

3D vectors are mathematical objects used to represent quantities with both magnitude (length) and direction in a three-dimensional space. They consist of three components: x, y, and z, and are often represented as (x, y, z).

## 2. How do you find the magnitude of a 3D vector?

The magnitude of a 3D vector can be found using the Pythagorean theorem, which states that the magnitude is equal to the square root of the sum of the squares of each component. In other words, the magnitude of a 3D vector (x, y, z) is equal to √(x² + y² + z²).

## 3. What is the difference between a vector and a point?

A vector represents a direction and magnitude in space, while a point represents a specific location in space. Vectors are often used to describe the relationship between points and can be used to calculate distances and angles between points.

## 4. How do you determine if a point lies on a line?

To determine if a point (x, y, z) lies on a line, you can use the parametric equation of a line, which is given by x = x₀ + at, y = y₀ + bt, and z = z₀ + ct, where (x₀, y₀, z₀) is a point on the line and a, b, and c are the direction ratios. If plugging in the coordinates of the point into this equation results in a true statement, then the point lies on the line.

## 5. What is the cross product of two 3D vectors?

The cross product of two 3D vectors is a vector that is perpendicular to both of the original vectors. It is calculated using the formula (a, b, c) × (d, e, f) = (bf - ce, cd - af, ae - bd). The magnitude of this cross product is equal to the product of the magnitudes of the two original vectors multiplied by the sine of the angle between them.