Question

Find vectors parallel to \(\mathbf{v}\) of the given length. $$\mathbf{v}=\langle 3,-2,6\rangle ; \text { length }=10$$

Short answer

Question: Find a vector parallel to \(\mathbf{v} = \langle 3, -2, 6 \rangle\) with a length of 10. Answer: \(\mathbf{w} = \langle 10, -20, 40 \rangle\)

Step-by-step solution

Find the magnitude of \(\mathbf{v}\)

To find the unit vector of \(\mathbf{v}=\langle 3,-2,6\rangle\), we need the magnitude of \(\mathbf{v}\). The magnitude is computed as follows: $$\lVert \mathbf{v} \rVert = \sqrt{3^2 + (-2)^2 + 6^2} = \sqrt{49} = 7$$

Compute the unit vector of \(\mathbf{v}\)

Now that we have the magnitude of \(\mathbf{v}\), we can compute the unit vector \(\mathbf{u}\) by dividing each component by the magnitude: $$\mathbf{u} = \frac{1}{\lVert \mathbf{v} \rVert}\mathbf{v} = \frac{1}{7}\langle 3, -2, 6\rangle = \left\langle \frac{3}{7}, -\frac{2}{7}, \frac{6}{7}\right\rangle$$

Find the vector parallel to \(\mathbf{v}\) with the desired length

Finally, we can find the vector parallel to \(\mathbf{v}\) with a length of 10 by multiplying the unit vector \(\mathbf{u}\) by 10: $$\mathbf{w} = 10\mathbf{u} = 10\left\langle \frac{3}{7}, -\frac{2}{7}, \frac{6}{7}\right\rangle = \langle 30\left(\frac{3}{7}\right), 30\left(-\frac{2}{7}\right), 30\left(\frac{6}{7}\right)\rangle = \langle 10, -20, 40\rangle$$ So, the vector parallel to \(\mathbf{v}\) with a length of 10 is \(\mathbf{w}=\langle 10, -20, 40\rangle\).

Key concepts

Magnitude of a Vector

Vectors are mathematical objects that possess both direction and magnitude. To understand vectors fully, calculating their magnitude is crucial. This is like figuring out how "long" the vector is.

The magnitude of a vector, often denoted by \(\lVert \mathbf{v} \rVert\), is evaluated by considering the vector's components. Suppose a vector \(\mathbf{v}\) has components \(a, b, c\). Its magnitude is calculated using the formula:

• \(\lVert \mathbf{v} \rVert = \sqrt{a^2 + b^2 + c^2}\),

which results from applying the Pythagorean Theorem in three dimensions.

A useful way to visualize this is to think of a point in space with coordinates given by the vector. The magnitude is the straight-line distance from the origin to this point. Thus, it lets us understand how big the vector is regardless of its direction.

In our example, to find the magnitude of \(\mathbf{v} = \langle 3, -2, 6 \rangle\), we computed \(\lVert \mathbf{v} \rVert = \sqrt{3^2 + (-2)^2 + 6^2} = \sqrt{49} = 7\). This tells us that vector \(\mathbf{v}\) has a size or length of 7 units.

Unit Vector Calculation

A unit vector is a vector that lists the direction of a given vector but has a magnitude of exactly 1. Unit vectors are especially useful for identifying directions and performing vector operations.

To find the unit vector of any given vector \(\mathbf{v}\), we scale down the original vector by its magnitude. Essentially, this involves dividing each component of the vector by the overall magnitude. The formula for a unit vector \(\mathbf{u}\)is:

• \(\mathbf{u} = \frac{1}{\lVert \mathbf{v} \rVert}\mathbf{v}\)

This operation ensures that \(\mathbf{u}\) maintains the direction of \(\mathbf{v}\) but is normalized to have no change in size other than simply being 1 unit long.

Taking an example: for the vector \(\langle 3, -2, 6 \rangle\) with a magnitude of 7, the unit vector becomes:

• \(\mathbf{u} = \frac{1}{7} \langle 3, -2, 6 \rangle = \left\langle \frac{3}{7}, -\frac{2}{7}, \frac{6}{7} \right\rangle\)

Notice how now any operation on this unit vector will lead only to changes in direction but will always maintain its normalized magnitude of 1.

Parallel Vectors

Parallel vectors are always pointing in the same or exactly opposite direction. They share a proportionality or scale relationship.

In other words, if you stretch or shrink a vector by some scaling factor, you end up with a parallel vector. Such scaling does not alter the direction relative to the original vector.

The simplest way to define parallel vectors is through multiplication of a vector's unit vector by any scalar length.

Finding Parallel Vectors

For instance, if a vector with a desired length has to be parallel to \(\mathbf{v}\), find its unit vector first, then scale this unit vector by the desired length. In our case, scaling the unit vector \(\mathbf{u} = \langle \frac{3}{7}, -\frac{2}{7}, \frac{6}{7} \rangle\) by 10 gives us the parallel vector \(\mathbf{w} = 10 \mathbf{u} = \langle 10, -20, 40 \rangle\).

Parallel vectors such as these will either overlap completely if positive or oppose each other if negative. Understanding parallel vectors is crucial for applications in physics, engineering, and computer graphics where directionality aligns movements or differentiates them based on context.