Problem 35 - Trigonometry

Question

For each vector, find \(\frac{1}{2} \mathbf{V},-\mathbf{V}\), and \(4 \mathbf{V}\). $$\mathbf{V}=\langle-3,7\rangle$$

Step-by-Step Solution

Answer

\(\frac{1}{2} \mathbf{V} = \langle -\frac{3}{2}, \frac{7}{2} \rangle\), \(-\mathbf{V} = \langle 3, -7 \rangle\), \(4\mathbf{V} = \langle -12, 28 \rangle\).

Step 1: Calculate \(\frac{1}{2} \mathbf{V}\)

To find the vector \(\frac{1}{2} \mathbf{V}\), we need to multiply each component of \(\mathbf{V}\) by \(\frac{1}{2}\). Therefore, \(\frac{1}{2} \mathbf{V} = \left\langle \frac{1}{2}(-3), \frac{1}{2}(7) \right\rangle = \langle -\frac{3}{2}, \frac{7}{2} \rangle\).

Step 2: Calculate \(-\mathbf{V}\)

To find the vector \(-\mathbf{V}\), we need to multiply each component of \(\mathbf{V}\) by \(-1\). Thus, \(-\mathbf{V} = \langle -1(-3), -1(7) \rangle = \langle 3, -7 \rangle\).

Step 3: Calculate \(4\mathbf{V}\)

To find the vector \(4\mathbf{V}\), we multiply each component of \(\mathbf{V}\) by 4. Therefore, \(4\mathbf{V} = \langle 4(-3), 4(7) \rangle = \langle -12, 28 \rangle\).

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential operation in vector mathematics that scales a vector by a constant value, known as the scalar. Think of the scalar as a dial that adjusts the vector’s magnitude while preserving its direction (unless the scalar is negative, in which case the direction reverses). This concept applies regardless of the size or direction of the original vector.

When we perform scalar multiplication, each component of the vector is multiplied by the scalar. For example, suppose we have a vector \(\mathbf{V} = \langle -3, 7 \rangle\) and want to perform the operation \(\frac{1}{2}\mathbf{V}\). In this case, our scalar is \(\frac{1}{2}\), and we multiply each component:

• The x-component: \(\frac{1}{2} \times -3 = -\frac{3}{2}\)
• The y-component: \(\frac{1}{2} \times 7 = \frac{7}{2}\)

Hence, the new vector is \(\langle -\frac{3}{2}, \frac{7}{2} \rangle\).

Scalar multiplication can also be thought of as stretching or shrinking a vector. A positive scalar maintains direction, while a negative scalar reverses it, which helps in operations like vector negation and further manipulation.

Vector Negation

Vector negation is a straightforward yet crucial concept. It involves changing the direction of a vector without affecting its magnitude, effectively flipping it by 180 degrees.

To negate a vector, you multiply each of its components by \(-1\). This process reverses the sign of each component, thus reversing the direction of the vector. For instance, consider vector \(\mathbf{V} = \langle -3, 7 \rangle\); to find \(-\mathbf{V}\), you follow these steps:

• Multiply the x-component: \(-1 \times -3 = 3\)
• Multiply the y-component: \(-1 \times 7 = -7\)

So, \(-\mathbf{V} = \langle 3, -7 \rangle\).

Flipping vectors is useful in many applications, especially when you need the opposite of a given direction. This is common in physics to illustrate forces acting in opposing directions or in navigation for plotting routes.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, bridges algebra and geometry by representing geometric figures through a coordinate system. In this system, vectors are described by their components, \(\mathbf{V} = \langle x, y \rangle\), within a plane.

Vectors simplify the representation of points and directions in a plane, allowing for concise mathematical descriptions and operations like addition, subtraction, and multiplication by scalars. With the vector \(\mathbf{V} = \langle -3, 7 \rangle\), each component corresponds to a specific part of the vector within a coordinate plane. This makes it easy to manipulate and understand.

• The x-component represents movement along the x-axis.
• The y-component indicates movement along the y-axis.

Combining these components gives us tools for tasks such as plotting points, determining line equations, and calculating distances.

In the context of vectors, operations like scalar multiplication and negation help in changing the shape, size, and direction of vectors. By understanding how vectors fit into the coordinate plane, these operations become intuitive tools for a variety of applications in both theoretical and applied contexts.