Problem 75 - Precalculus with Limits

Question

Einding the Component Form of a Vector In Exercises \(75-78\) , find the component form of the sum of u and v with direction angles \(\theta_{\text { u }}\) and \(\theta_{v}\) . $$\begin{array}{ll}{\text { Magnitude }} & {\text { Angle }} \\\ {\|\mathbf{u}\|=5} & {\theta_{\mathrm{u}}=0^{\circ}} \\ {\|\mathbf{v}\|=5} & {\theta_{\mathrm{v}}=90^{\circ}}\end{array}$$

Step-by-Step Solution

Answer

The component form of the sum of vectors u and v is (5, 5).

Step 1: Convert Polar to Cartesian Coordinates

Before proceeding with adding vectors, convert the given polar coordinates into Cartesian coordinates. The conversion formulas are as follows: \(x = r*cos(\Theta)\), \(y = r*sin(\Theta)\) (where r is the magnitude and \(\Theta\) is the angle of the vector). Apply the formula and calculate the respective x, y coordinates for vectors u and v.\n\nFor vector u: \(x_u = 5*cos(0^\circ) = 5\), \(y_u = 5*sin(0^\circ) = 0\) hence \(u=(5, 0)\).\n\nFor vector v: \(x_v = 5*cos(90^\circ) = 0\), \(y_v = 5*sin(90^\circ) = 5\) hence \(v=(0, 5)\).

Step 2: Adding the Vectors

Now, add both vectors u and v. The addition of two vectors entails adding their corresponding components.\n\nTherefore, \(u+v\) will be the vector whose x-coordinate is \(x_u + x_v\) and whose y-coordinate is \(y_u + y_v\). Calculate the x and y components of the vector \(u+v\).\n\nFor x: \(x_u + x_v = 5 + 0 = 5\)\n\nFor y: \(y_u + y_v = 0 + 5 = 5\).\n\nThus, the component form of the sum of vectors u and v is (5, 5).

Key Concepts

Vector Addition

The process of vector addition often seems tricky at first glance, but it is merely a way of combining two vectors to find a resultant vector. Imagine you're taking a walk: you first stroll 5 blocks east (let's call this vector u), then 5 blocks north (vector v). The path from your starting point directly to your location, skipping all the twists and turns, is the resultant vector (u+v).

In mathematical terms, vector addition involves combining the corresponding components of two vectors. This means adding the x-coordinates of the vectors to get the new x-coordinate, and doing the same with the y-coordinates for the resultant vector's y-coordinate. For the vectors in our exercise, you'd calculate it as follows: For x: \(x_u + x_v = 5 + 0 = 5\)For y: \(y_u + y_v = 0 + 5 = 5\).The component form of the sum of vectors u and v intuitively is the diagonal in a square, which in this case is (5, 5).

To visualize it better, drawing the vectors on a grid can show how they add up head-to-tail, and the resultant vector will stretch from the tail of the first vector to the head of the second.

Polar to Cartesian Coordinates

In the realm of mathematics, we often have to switch between different systems of coordinates, and two of the most common are polar and Cartesian coordinates. Polar coordinates gauge the location of a point based on its angle and distance from the origin, whereas Cartesian coordinates use two perpendicular axes (commonly x and y) to define a point's location.

To convert polar coordinates (which consist of a radius and an angle, such as \( (r, \theta) \)) to Cartesian coordinates (the familiar (x, y)), we use two simple equations: \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \).In the exercise, these formulas were applied to turn the magnitudes and direction angles of vectors u and v into recognizable x and y coordinates:
For vector u with \( \theta_u = 0^\circ \), we find \( x_u = 5 \cdot \cos(0^\circ) = 5 \) and \( y_u = 5 \cdot \sin(0^\circ) = 0 \), which results in Cartesian coordinates of (5, 0).
Similarly, for vector v with \( \theta_v = 90^\circ \), calculation yields \( x_v = 5 \cdot \cos(90^\circ) = 0 \) and \( y_v = 5 \cdot \sin(90^\circ) = 5 \), leading to Cartesian coordinates of (0, 5).

These equations allow for seamless transitions between coordinate systems, making it easier to work on a variety of mathematical problems.

Direction Angles

Direction angles are vital when it comes to navigating the world of vectors, as they provide a way of describing the directional bearing of a vector from a standardized point, usually the positive x-axis. The angle is measured in degrees or radians and signifies the rotation required to align the x-axis to the vector direction.

For instance, a direction angle of \(0^\circ\) indicates no rotation — the vector points right along the positive x-axis. A direction angle of \(90^\circ\) signifies the vector is pointing straight up along the positive y-axis. In our textbook problem, vector u with \(\theta_u = 0^\circ\) points directly along the positive x-axis; vector v with \(\theta_v = 90^\circ\) aligns with the positive y-axis.

These angles are not just abstract concepts but are practical tools for converting between polar and Cartesian coordinate systems, as we have seen in the exercise. They help to visualize vectors graphically and to maneuver them analytically with precision and ease. Understanding direction angles is crucial for anyone delving into vector analysis, physics, or any field that involves vector quantities.