Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\). $$\mathbf{v}=\frac{1}{2}(3 \mathbf{u}+\mathbf{w})$$

Step-by-Step Solution

Answer

The component form of \( \mathbf{v} \) is \(3.5\mathbf{i} - 0.5\mathbf{j}\). When the vectors are sketched geometrically, vector \( \mathbf{v} \) lies between vectors \( \mathbf{u} \) and \( \mathbf{w} \).

Step 1: Identify the vectors

Identify the given vectors. Here, vectors are: \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\)

Step 2: Multiply Vector \( \mathbf{u} \) by 3

The vector \(\mathbf{u}\) should be multiplied by 3. The operation results to \(3\mathbf{u}=6 \mathbf{i} - 3 \mathbf{j}\)

Step 3: Add vector \( \mathbf{w} \) to \( 3\mathbf{u} \)

Add vectors \(3\mathbf{u}\) and \( \mathbf{w} \) together: \(3\mathbf{u} + \mathbf{w} = (6 \mathbf{i} - 3 \mathbf{j}) + (\mathbf{i}+2 \mathbf{j}) = 7 \mathbf{i} - \mathbf{j}\)

Step 4: Determine Vector \( \mathbf{v} \)

Substitute the result into the expression for vector \(\mathbf{v}\). This means the combination of \(3\mathbf{u} + \mathbf{w}\) should be divided by 2, that results to \(\mathbf{v}=\frac{1}{2}(3 \mathbf{u}+\mathbf{w}) = \frac{1}{2}(7 \mathbf{i}-\mathbf{j}) = 3.5\mathbf{i} - 0.5\mathbf{j}\)

Step 5: Sketch the vectors

Sketch the vectors \( \mathbf{u} \), \( \mathbf{w} \), and \( \mathbf{v} \) geometrically, using the i and j components as coordinates in a standard Cartesian plane. Vector \(\mathbf{u}\) is located at (2, -1), \(\mathbf{w}\) is at (1, 2), and \(\mathbf{v}\) is at (3.5, -0.5). The tail of each vector starts at the origin (0,0) and the head ends at the respective coordinates. The vector \(\mathbf{v}\) should lie between \(\mathbf{u}\) and \(\mathbf{w}\) since it is a combination of the two.

Key Concepts

Vector Addition

Vector addition is a fundamental operation in algebra that involves combining two or more vectors to produce a new vector, referred to as the resultant vector. The process of vector addition follows specific rules depending on the way it is carried out: graphically or algebraically.

When adding vectors graphically, we align them head to tail in sequence and draw the resultant vector from the tail of the first vector to the head of the last. Algebraically, we can simply add the corresponding components – that is, the i (horizontal) components are added, and the j (vertical) components added separately. For example, when adding \(\mathbf{u}=2\mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2\mathbf{j}\), the resultant vector's components are calculated by \(\mathbf{u} + \mathbf{w} = (2\mathbf{i}+\mathbf{i}) + (-\mathbf{j}+2\mathbf{j}) = 3\mathbf{i} + \mathbf{j}\).

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction if the scalar is positive. The mathematical representation can be simplified by multiplying each component of the vector by the scalar.

For instance, the exercise demonstrates scalar multiplication with the vector \(\mathbf{u}=2\mathbf{i}-\mathbf{j}\) being multiplied by the scalar 3. The operation results in \(3\mathbf{u}=3(2\mathbf{i}-\mathbf{j})=6\mathbf{i}-3\mathbf{j}\). Each component of the original vector \(\mathbf{u}\) has been scaled by the factor of 3. This is a crucial operation for changing the size of vectors in various applications including physics and engineering.

Sketching Vectors Geometrically

Sketching vectors geometrically on a Cartesian plane allows us to visually interpret the direction and magnitude of vectors. The Cartesian plane is a two-dimensional grid where each point can be specified by its x (horizontal) and y (vertical) components.

To sketch a vector, start at the origin (0,0) and draw a line to the point represented by its components. For instance, the vector \(\mathbf{u}\) would be drawn from (0,0) to (2,-1), and similarly, \(\mathbf{w}\) from (0,0) to (1,2). The sketch is particularly helpful for visualizing vector operations such as addition, as shown in the problem above where the resultant vector \(\mathbf{v}\) lies between \(\mathbf{u}\) and \(\mathbf{w}\), reflecting its composition.

Cartesian Plane

The Cartesian plane is a crucial concept in algebra and geometry, consisting of a two-dimensional surface defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Together, these axes divide the plane into four quadrants, where each point is identified by an ordered pair of coordinates: \(x,y\).

The location of points on this plane corresponds to the components of vectors. For example, a vector with components \(2\mathbf{i} - \mathbf{j}\) might be represented as a point or arrow starting from the origin (0, 0) and extending to the point (2, -1). This representations allows us to perform vector operations geometrically and helps in understanding concepts like vector addition and scalar multiplication in a visual and intuitive manner.