Question

Find a parametric equation of the line through \(\mathbf{X}_{0}\) in the direction of \(\mathbf{U}\). (a) \(\mathbf{X}_{0}=(1,2,-3,1), \quad \mathbf{U}=(3,4,5,-4)\) (b) \(\mathbf{X}_{0}=(2,0,-1,2,4), \quad \mathbf{U}=(-1,0,1,3,2)\) (c) \(\mathbf{X}_{0}=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{4},-\frac{1}{4}\right), \quad \mathbf{U}=\left(\frac{1}{3},-\frac{1}{6}, \frac{1}{6},-\frac{1}{3}\right)\)

Short answer

Question: Determine the parametric equation of the line through the given point and in the direction of the given vector. (a) Point: (1,2,-3,1); Direction vector: (3,4,5,-4) (b) Point: (2,0,-1,2,4); Direction vector: (-1,0,1,3,2) (c) Point: (-1/2, 1/2, 1/4, -1/4); Direction vector: (1/3, -1/6, 1/6, -1/3) Answer: (a) The parametric equation of the line is: \(\mathbf{X}(t) = (1+3t, 2+4t, -3+5t, 1-4t)\) (b) The parametric equation of the line is: \(\mathbf{X}(t) = (2-t, t, -1+t, 2+3t, 4+2t)\) (c) The parametric equation of the line is: \(\mathbf{X}(t) = \left(-\frac{1}{2}+\frac{1}{3}t, \frac{1}{2}-\frac{1}{6}t, \frac{1}{4}+\frac{1}{6}t, -\frac{1}{4}-\frac{1}{3}t\right)\)

Step-by-step solution

Write the general parametric representation of a line

The formula for the parametric equation of a line is given by: \(\mathbf{X}(t) = \mathbf{X}_{0} + t\mathbf{U}\).

Plug in the given values

Now, we can substitute the values \(\mathbf{X}_{0}=(1,2,-3,1)\) and \(\mathbf{U}=(3,4,5,-4)\) into the equation: \(\mathbf{X}(t) = (1,2,-3,1) + t(3,4,5,-4)\).

Simplify the equation

After substituting, the equation simplifies to: \(\mathbf{X}(t) = (1+3t, 2+4t, -3+5t, 1-4t)\). (b)

Write the general parametric representation of a line

The formula for the parametric equation of a line is given by: \(\mathbf{X}(t) = \mathbf{X}_{0} + t\mathbf{U}\).

Plug in the given values

Now, we can substitute the values \(\mathbf{X}_{0}=(2,0,-1,2,4)\) and \(\mathbf{U}=(-1,0,1,3,2)\) into the equation: \(\mathbf{X}(t) = (2,0,-1,2,4) + t(-1,0,1,3,2)\).

Simplify the equation

After substituting, the equation simplifies to: \(\mathbf{X}(t) = (2-t, t, -1+t, 2+3t, 4+2t)\). (c)

Write the general parametric representation of a line

The formula for the parametric equation of a line is given by: \(\mathbf{X}(t) = \mathbf{X}_{0} + t\mathbf{U}\).

Plug in the given values

Now, we can substitute the values \(\mathbf{X}_{0}=\left(-\frac{1}{2}, \frac{1}{2},\frac{1}{4},-\frac{1}{4}\right)\) and \(\mathbf{U}=\left(\frac{1}{3},-\frac{1}{6}, \frac{1}{6},-\frac{1}{3}\right)\) into the equation: \(\mathbf{X}(t) = \left(-\frac{1}{2}, \frac{1}{2},\frac{1}{4},-\frac{1}{4}\right) + t\left(\frac{1}{3},-\frac{1}{6}, \frac{1}{6},-\frac{1}{3}\right)\).

Simplify the equation

After substituting, the equation simplifies to: \(\mathbf{X}(t) = \left(-\frac{1}{2}+\frac{1}{3}t, \frac{1}{2}-\frac{1}{6}t, \frac{1}{4}+\frac{1}{6}t, -\frac{1}{4}-\frac{1}{3}t\right)\).

Key concepts

Vectors

Vectors are fundamental elements in mathematics and physics. They represent both magnitude and direction, such as velocity in physics or displacement in geometry. A vector can be seen as an arrow in space, defined by its coordinates in a Cartesian plane or space. For example, a vector \( \mathbf{U} = (3, 4, 5, -4) \) is a four-dimensional vector, where each number is a component in a different dimension.
One of the key characteristics of vectors is that they can be added together to form new vectors. This property is crucial for defining lines and planes in spaces, as seen in parametric equations.

• Addition: Vectors are added component-wise. If you have vectors \( \mathbf{A} = (a_1, a_2, a_3) \) and \( \mathbf{B} = (b_1, b_2, b_3) \), their sum is \( \mathbf{A} + \mathbf{B} = (a_1+b_1, a_2+b_2, a_3+b_3) \).
• Scalar Multiplication: If you multiply a vector by a scalar, you simply multiply each component of the vector by the scalar. For instance, multiplying \( \mathbf{U} \) by \( t \) will scale the original vector.

Understanding these basic operations is essential for working with vectors in parametric equations.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors and linear transformations. It provides the tools necessary to handle multiple variables simultaneously and is deeply connected with parametric equations, like the ones used to describe lines in space.
A line can be expressed literally as a collection of vectors that satisfy a linear equation. Parametric equations are able to describe such lines using vectors, as seen when solving the exercise using \( \mathbf{X}(t) = \mathbf{X}_0 + t\mathbf{U} \).

• Parametric Representation: This form is a fundamental concept in linear algebra and is used to solve problems that involve lines in higher dimensions.
• Systems of Equations: These methods are used to find where lines intersect or to find the equation of a line given points and directions in space.

Grasping linear algebra allows one to handle vectors not just in two-dimensional planes, but in multi-dimensional spaces.

Directional Vectors

Directional vectors specify a direction in space and are critical when defining lines. These vectors show "where to go" from one point to another. In a parametric equation of a line, like \( \mathbf{X}(t) = \mathbf{X}_0 + t\mathbf{U} \), the vector \( \mathbf{U} \) acts as the directional vector.
Directional vectors are widely used due to their effectiveness in representing movement, force, or any other vector quantity in a specific direction. When constructing parametric equations:

• The directional vector defines consistency in the line's direction, ensuring the line extends infinitely in the specified direction.
• Changes in the magnitude of the direction vector determine the scaling of the line's extension, altering how 'quickly' the line extends.

These characteristics make directional vectors fundamental in applications like computer graphics, physics simulations, and route planning.