Problem 19 - Advanced Engineering Mathematics

Question

Find the terminal point of the vector \(P_{1} P_{2}=4 \mathbf{i}+8 \mathbf{j}\) if its initial point is \((-3,10)\).

Step-by-Step Solution

Answer

The terminal point is \((1, 18)\).

Step 1: Understanding the Problem

We have an initial point of the vector, which is \((-3, 10)\), and we need to find the terminal point using the given vector \(P_{1}P_{2} = 4\mathbf{i} + 8\mathbf{j}\). This vector indicates the change in the x-coordinate and the y-coordinate between the initial and terminal points.

Step 2: Determining Change in Coordinates

The vector \(\mathbf{v} = 4\mathbf{i} + 8\mathbf{j}\) tells us the movement from the initial point. The \(i\) component represents a change of \(4\) in the x-direction, and the \(j\) component represents a change of \(8\) in the y-direction.

Step 3: Calculating the Terminal Point Coordinates

To find the terminal point, we add the components of the vector to the coordinates of the initial point. The x-coordinate of the terminal point will be \(-3 + 4 = 1\), and the y-coordinate will be \(10 + 8 = 18\).

Step 4: Writing the Terminal Point

Using the computed changes in coordinates, the terminal point of the vector \(P_{1}P_{2}\) is \((1, 18)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use a coordinate system to examine and interpret geometric properties and relationships.

• In coordinate geometry, points are expressed as coordinates, typically \((x, y)\) in two-dimensional space.
• These coordinates help us determine distances, slopes, and other characteristics by using algebra and calculus.
• With vectors, coordinate geometry allows us to explore transformations and movements in a plane.

In this particular task, we're given an initial point and a vector. Using coordinate geometry, we translate from that initial point to find the terminal point of the vector, illustrating how coordinate geometry simplifies solving for position changes in space.

Vectors and Scalars

Vectors and scalars are fundamental concepts in physics and mathematics that describe different types of quantities.

• A scalar is a quantity that only has magnitude. For example, temperature, mass, and length.
• A vector, on the other hand, not only has magnitude but also direction. This makes vectors particularly useful in representing physical entities like force, velocity, and displacement.

In the exercise, \(4\mathbf{i} + 8\mathbf{j}\) is a vector showing movement in two dimensions. Here, \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors pointing in the direction of the x and y axes respectively, and the numbers 4 and 8 represent how far the point moves along those axes. Understanding the difference between scalars and vectors is key to interpreting and using them appropriately in problems involving directions and magnitudes.

Vector Addition

Vector addition is a crucial operation that allows us to determine resultant vectors, especially when considering multiple movements or forces.

• When adding vectors, you consider both their magnitude and direction.
• Geometrically, vector addition can often be visualized using the "tip-to-tail" method.
• However, analytically, vector addition involves adding together their components.

In our solution, we start with the initial point \((-3, 10)\) and use vector addition to incorporate the changes from \(4\mathbf{i} + 8\mathbf{j}\). This means adding 4 to the x-coordinate and 8 to the y-coordinate of the initial point, giving us \((1, 18)\) as the terminal point. Recognizing the straightforwardness of vector addition is key to solving many problems in coordinate geometry.