Question

State the coordinates of the point \(\mathrm{P}\) if its position vector is given as (a) \(3 \boldsymbol{i}-7 \boldsymbol{j}\), (b) \(-4 \boldsymbol{i}\), (c) \(-0.5 \boldsymbol{i}+13 \boldsymbol{j}\), (d) \(a \boldsymbol{i}+b \boldsymbol{j}\).

Short answer

Question: Determine the coordinates of point P for each position vector: (a) \(3\boldsymbol{i} - 7\boldsymbol{j}\) (b) \(-4\boldsymbol{i}\) (c) \(-0.5\boldsymbol{i} + 13\boldsymbol{j}\) (d) \(a\boldsymbol{i} + b\boldsymbol{j}\) Answer: (a) (3, -7) (b) (-4, 0) (c) (-0.5, 13) (d) (a, b)

Step-by-step solution

(a) Determine the coordinates of P for the position vector \(3\boldsymbol{i} - 7\boldsymbol{j}\)

Find the coefficients of \(\boldsymbol{i}\) and \(\boldsymbol{j}\). The coefficient of \(\boldsymbol{i}\) is 3, and the coefficient of \(\boldsymbol{j}\) is -7. Thus, the coordinates of point P are (3, -7).

(b) Determine the coordinates of P for the position vector \(-4\boldsymbol{i}\)

Find the coefficients of \(\boldsymbol{i}\) and \(\boldsymbol{j}\). The coefficient of \(\boldsymbol{i}\) is -4, and the coefficient of \(\boldsymbol{j}\) is not provided, which means it is 0. Therefore, the coordinates of point P are (-4, 0).

(c) Determine the coordinates of P for the position vector \(-0.5\boldsymbol{i} + 13\boldsymbol{j}\)

Find the coefficients of \(\boldsymbol{i}\) and \(\boldsymbol{j}\). The coefficient of \(\boldsymbol{i}\) is -0.5, and the coefficient of \(\boldsymbol{j}\) is 13. Hence, the coordinates of point P are (-0.5, 13).

(d) Determine the coordinates of P for the position vector \(a\boldsymbol{i} + b\boldsymbol{j}\)

Find the coefficients of \(\boldsymbol{i}\) and \(\boldsymbol{j}\). The coefficient of \(\boldsymbol{i}\) is a, and the coefficient of \(\boldsymbol{j}\) is b. So, the coordinates of point P are (a, b).

Key concepts

Coordinates in Mathematics

Coordinates are a set of values used to determine the position of a point in a space of a given dimension. In a two-dimensional space, commonly referred to as a plane, these values are known as Cartesian coordinates and are denoted as a pair \( (x, y) \), where \( x \) represents the horizontal component, and \( y \) represents the vertical component.

To find the coordinates of a point from its position vector—as in the exercise, where the position vectors are given in terms of basis vectors \( \boldsymbol{i} \) and \( \boldsymbol{j} \)—one must extract the coefficients that multiply these unit vectors. The coefficient in front of \( \boldsymbol{i} \) gives us the \( x \) value, and the coefficient in front of \( \boldsymbol{j} \) gives us the \( y \) value. If a coefficient is not explicitly given, it is understood to be zero. This knowledge assists students in visualizing geometric representations of algebraic expressions and is fundamental in fields ranging from physics to computer graphics.

Vectors in Engineering

In engineering, vectors are vital for representing quantities that have both magnitude and direction, such as force, velocity, or acceleration. A vector's representation often uses a combination of basis vectors—such as \( \boldsymbol{i} \) and \( \boldsymbol{j} \) for two-dimensional vectors in Cartesian coordinates—which indicate direction along the x and y axes, respectively.

The position vector, for instance, identifies the location of a point relative to an origin. In the exercise solutions, position vectors are given in the format of \( a\boldsymbol{i} + b\boldsymbol{j} \) and are utilized to define the coordinates of a point \( P \) in a plane. This concept is not only central to engineering problem-solving but also to understanding how to navigate planes and three-dimensional spaces, such as in robotics or aeronautics.

Mathematical Coefficients

Mathematical coefficients are the numerical or algebraic factors that multiply the variables or basis vectors in an expression. They provide essential information about the scale or proportion of these variables in relation to an equation or vector.

In the context of vectors, as seen in the exercise, the coefficients of \( \boldsymbol{i} \) and \( \boldsymbol{j} \) effectively size and orient the vector in the xy-plane. For example, if a vector is represented as \( 3\boldsymbol{i} - 7\boldsymbol{j} \), the coefficient \( 3 \) indicates that the vector extends 3 units in the positive direction of the x-axis, while the coefficient \( -7 \) indicates a 7 unit extension in the negative direction of the y-axis. Recognizing the significance of coefficients is crucial for understanding linear combinations in vector spaces and for the manipulation of various mathematical and engineering constructs.