Problem 49 - Precalculus

Question

Express the vector in terms of unit vectors i and \(j\) $$(-1,0)$$

Step-by-Step Solution

Answer

The vector is \(-\mathbf{i}\) in terms of unit vectors.

Step 1: Identify the vector components

The vector given is \((-1, 0)\), which means it has an \(x\)-component of \(-1\) and a \(y\)-component of \(0\).

Step 2: Understand unit vectors i and j

The unit vector \(\mathbf{i}\) is defined as \((1, 0)\), and the unit vector \(\mathbf{j}\) is defined as \((0, 1)\). These represent the directions in the \(x\) and \(y\) axes, respectively.

Step 3: Express the vector using i

Since the \(x\)-component of the vector is \(-1\), we multiply the unit vector \(\mathbf{i}\) by \(-1\). Thus, the component along \(\mathbf{i}\) is \(-1 \cdot \mathbf{i}\).

Step 4: Express the vector using j

Since the \(y\)-component is \(0\), multiplying the unit vector \(\mathbf{j}\) by \(0\) gives us \(0 \cdot \mathbf{j}\), meaning there is no component along \(\mathbf{j}\).

Step 5: Combine the components

Add the contributions from \(\mathbf{i}\) and \(\mathbf{j}\) to form the vector in terms of unit vectors: \(-1 \cdot \mathbf{i} + 0 \cdot \mathbf{j}\). This simplifies to \(-\mathbf{i}\).

Key Concepts

Unit Vector

In the study of vectors, a unit vector is a fundamental concept. A unit vector is essentially a vector that has a magnitude of exactly one unit. It serves a fundamental role in defining directions without influencing the length of a vector. When using unit vectors, you can define the direction of a line in space strictly, as unit vectors have no effect on scale.

In two-dimensional spaces, the unit vectors are frequently denoted as \(\mathbf{i}\) and \(\mathbf{j}\). These reflect the principal direction along the axes:

• Unit vector \(\mathbf{i}\) is directed along the positive \(x\)-axis, represented as \((1, 0)\).
• Unit vector \(\mathbf{j}\) is directed along the positive \(y\)-axis, represented as \((0, 1)\).

These vectors primarily indicate the direction without impacting the magnitude of your vector, making them extremely useful for defining and working with directional vectors.

Vector Components

Visualizing a vector as being composed of different parts helps in understanding its true essence and behavior. Each vector can be broken down into what are known as its components. These are parallel to the coordinate system's principal directions.

To express a vector, we focus on:

• Its \(x\)-component, which shows the extent of the vector in the horizontal direction.
• Its \(y\)-component, which shows the extent of the vector in the vertical direction.

For example, in the vector \((-1, 0)\), the \(x\)-component is \(-1\) and the \(y\)-component is \(0\). This indicates the vector is negatively positioned along the \(x\)-axis while it has no vertical movement. Each component can be multiplied by the unit vector corresponding to its direction to express the overall vector.

Vector Notation

Vector notation provides a standard way to articulate vectors in terms of their components along chosen directions. This way of representation is clean, precise, and widely accepted in mathematics and physics.

Typically, vectors are expressed as a linear combination of unit vectors. This means:

• We multiply each vector component by its corresponding unit vector.
• These products are then summed to form the complete vector.

For instance, the vector \((-1, 0)\) can be denoted as \(-1 \cdot \mathbf{i} + 0 \cdot \mathbf{j}\), which readily simplifies to \(-\mathbf{i}\). Using vector notation is essential in simplifying complex vector expressions and performing vector-related operations efficiently. It clearly emphasizes both the direction and the magnitude of the vector components.