Problem 61 - Precalculus with Limits

Question

In Exercises 57-62, find the component form of \(\small{\mathbf{v}}\) and sketchthe specified vector operations geometrically, where \(\small{\mathbf{u} = 2\mathbf{i} - \mathbf{j}}\), and \(\small{\mathbf{w} = \mathbf{i} + 2\mathbf{j}}\). \(\small{\mathbf{v} = \frac{1}{2} (3\mathbf{u} + \mathbf{w})}\)

Step-by-Step Solution

Answer

The component form of vector \(\mathbf{v}\) is (7/2, -1/2).

Step 1: Express Vectors in Component Form

To start, express both vector \(\mathbf{u}\) and \(\mathbf{w}\) in component form. Vector \(\mathbf{u}\) is \(2\mathbf{i} - \mathbf{j}\), so its components are (2, -1). Vector \(\mathbf{w}\) is \(\mathbf{i} + 2\mathbf{j}\), so its components are (1, 2) .

Step 2: Perform the Vector Operations

Now, perform the addition and scalar multiplication operations specified by the equation for \(\mathbf{v}\). It can be rewritten as: \(\mathbf{v} = \frac{1}{2}(3\mathbf{u} + \mathbf{w}) = \frac{1}{2} ((3 * 2, 3 * -1) + (1, 2)) = \frac{1}{2} ((6, -3) + (1, 2)) = \frac{1}{2}(7, -1)\).

Step 3: Simplify the Vector

Lastly, divide each vector component by 2 to complete the scalar division operation. This yields \(\mathbf{v} = (7/2, -1/2) \) .

Key Concepts

Component Form of Vectors

Understanding the component form of vectors is fundamental in precalculus and physics. A vector in component form is represented by an ordered pair (or triplet for three-dimensional vectors) that shows the vector's magnitude along each axis.

In two dimensions, a vector \( \mathbf{v} \) is represented as \( (v_x, v_y) \), where \( v_x \) and \( v_y \) are the horizontal and vertical components, respectively. These components correspond to how far the vector moves in the x-direction and y-direction.

For example, the vector \( \mathbf{u} = 2\mathbf{i} - \mathbf{j} \) given in the exercise has components (2, -1). Here, \( 2\mathbf{i} \) represents the vector's movement two units to the right along the x-axis, and \( -\mathbf{j} \) represents the movement one unit down along the y-axis. This component form simplifies calculations involving vectors.

Vector Addition

Vector addition follows simple geometric rules that can be visualized as connecting the tail of one vector to the head of another.

Using the components, vector addition can be done algebraically by adding the corresponding components of each vector together. If we have two vectors \( \mathbf{a} = (a_x, a_y) \) and \( \mathbf{b} = (b_x, b_y) \) in two dimensions, their sum \( \mathbf{c} = \mathbf{a} + \mathbf{b} \) has components \( (a_x + b_x, a_y + b_y) \). This gives us a new vector whose x-component is the sum of the x-components of \( \mathbf{a} \) and \( \mathbf{b} \) and whose y-component is the sum of their y-components.

Following the exercise, by adding the vectors \( \mathbf{u} \) and \( \mathbf{w} \) after scalar multiplication, \( (6, -3) + (1, 2) \) gives us the components of the resulting vector \( (7, -1) \) before the final scalar division.

Scalar Multiplication of Vectors

The concept of scalar multiplication involves multiplying a vector by a real number (called a scalar). This operation scales the vector's magnitude without changing its direction.

In component form, scalar multiplication is performed by multiplying each component of the vector by the scalar. For example, if we have a vector \( \mathbf{v} = (v_x, v_y) \) and a scalar \( k \) , the product is \( k\mathbf{v} = (kv_x, kv_y) \).

In our exercise, the vector \( \mathbf{v} \) is obtained by first multiplying the vector \( \mathbf{u} \) by 3 (scalar) and then multiplying the resultant vector by \( \frac{1}{2} \) (another scalar). This sequential scalar multiplication illustrates how vectors can be scaled and combined through algebraic operations.