Question

Find vectors parallel to \(\mathbf{v}\) of the given length. $$\mathbf{v}=\overrightarrow{P Q} \text { with } P(1,0,1) \text { and } Q(2,-1,1) ; \text { length }=3$$

Short answer

2. Calculate the magnitude of the vector \(\overrightarrow{PQ}\) obtained in step 1. 3. What is the unit vector in the direction of \(\overrightarrow{PQ}\)? 4. What is the vector \(\mathbf{w}\) parallel to \(\mathbf{v}\) with the given length of 3?

Step-by-step solution

Find the vector \(\mathbf{v}=\overrightarrow{PQ}\)

Given points P(1,0,1) and Q(2,-1,1), calculate \(\overrightarrow{PQ}\) by subtracting the coordinates of P from the coordinates of Q, i.e., $$\overrightarrow{PQ} = (Q_x - P_x, Q_y - P_y, Q_z - P_z)$$ Substitute the coordinates of P and Q and calculate the components of \(\overrightarrow{PQ}\).

Calculate the magnitude of \(\overrightarrow{PQ}\)

To find the magnitude (length) of vector \(\overrightarrow{PQ}\), use the formula $$||\overrightarrow{PQ}|| = \sqrt{(Q_x - P_x)^2 + (Q_y - P_y)^2 + (Q_z - P_z)^2}$$ Substitute the components of \(\overrightarrow{PQ}\) and calculate the magnitude.

Calculate the unit vector in the direction of \(\overrightarrow{PQ}\)

To find the unit vector in the same direction as \(\overrightarrow{PQ}\), divide each component of \(\overrightarrow{PQ}\) by the magnitude calculated in step 2, $$\widehat{PQ} = \frac{\overrightarrow{PQ}}{||\overrightarrow{PQ}||}$$ Calculate the components of the unit vector \(\widehat{PQ}\).

Multiply the unit vector by the desired length

To obtain the vector parallel to \(\overrightarrow{PQ}\) with the desired length of 3, multiply the unit vector \(\widehat{PQ}\) by the length, $$\mathbf{w} = 3 \times \widehat{PQ}$$ Calculate the components of the vector \(\mathbf{w}\) which is parallel to \(\mathbf{v}\) and has a length of 3. The vector \(\mathbf{w}\) is the desired vector parallel to \(\mathbf{v}\) with the given length.

Key concepts

Vector Algebra

Vector algebra is a branch of mathematics that deals with vectors, which are quantities having both magnitude and direction. Unlike scalar quantities, which have only magnitude, vectors incorporate direction, making them vital for representing physical quantities such as force, velocity, and acceleration.

In vector algebra, we perform operations such as addition, subtraction, and scalar multiplication, which obey certain rules and properties analogous to regular algebra. For example, vector addition is commutative, meaning that the order in which vectors are added does not affect the result: \( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \).

When working with coordinate systems, vectors can be represented in terms of their components along the axes. For instance, in a three-dimensional coordinate system, a vector \( \mathbf{v} \) can be represented as \( \mathbf{v} = (v_x, v_y, v_z) \). This representation allows us to perform vector arithmetic by manipulating each component individually, leading to a clear geometric interpretation of the operations.

Unit Vectors

Unit vectors play a critical role in vector algebra, serving as the fundamental building blocks to describe directions in space. A unit vector is a vector with a magnitude of one, thus it only conveys direction. In essence, unit vectors are used to express any vector as a scaled version of a certain direction.

To obtain a unit vector, you take any vector and divide it by its magnitude, effectively normalizing it. Mathematically, for a vector \( \mathbf{v} \), its corresponding unit vector \( \widehat{v} \) is calculated as \( \widehat{v} = \frac{\mathbf{v}}{||\mathbf{v}||} \), where \( ||\mathbf{v}|| \) is the magnitude of vector \( \mathbf{v} \).

This process of normalization is vital when you want to keep the direction of a vector but change its magnitude to something else, such as creating vectors in the same direction with a predefined length, as seen in the textbook exercise above.

Magnitude of a Vector

The magnitude of a vector, often called its length or norm, is a measure of how long the vector is, irrespective of its direction. The concept of magnitude is essential in understanding vectors as it quantifies the size of a vector in a scalar form.

The magnitude of a vector in three-dimensional space \( \mathbf{v} = (v_x, v_y, v_z) \) is given by the formula \( ||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2 + v_z^2} \). This is analogous to finding the length of the hypotenuse of a right-angled triangle using Pythagoras' theorem, but extended into three dimensions.

Knowing how to calculate the magnitude is particularly useful when trying to find a vector of a specific length that is parallel to an existing vector—as illustrated in the textbook exercise. By scaling the unit vector in the given direction by the desired length, you can construct a new vector with the required properties.