Question

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$$

Short answer

Based on the step-by-step solution provided, the short answer to the problem is: By breaking down the vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) into their respective components and using the definition of vector addition, we have proved that the property of associativity holds: $$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$$ Additionally, we can draw a sketch to illustrate this property geometrically, showing that point D remains the same in both the left-hand side and the right-hand side additions. Therefore, the addition of vectors is associative.

Step-by-step solution

Write down the given vectors in terms of their components

Let the vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) be given as follows: \(\mathbf{u} = u_x \hat{i}+ u_y \hat{j}\) \(\mathbf{v} = v_x \hat{i}+ v_y \hat{j}\) \(\mathbf{w} = w_x \hat{i}+ w_y \hat{j}\)

Prove associativity using components

First, let's calculate the left-hand side of the equation: \((\mathbf{u}+\mathbf{v})+\mathbf{w} = (u_x \hat{i}+ u_y \hat{j} + v_x \hat{i}+ v_y \hat{j}) + (w_x \hat{i}+ w_y \hat{j})\) Group the \(\hat{i}\) and \(\hat{j}\) components together: \(= (u_x + v_x) \hat{i}+(u_y + v_y) \hat{j} + w_x \hat{i}+ w_y \hat{j}\) Now add the components with \(\mathbf{w}\): \(= (u_x + v_x + w_x) \hat{i}+(u_y + v_y + w_y) \hat{j}\) Now let's calculate the right-hand side of the equation: \(\mathbf{u}+(\mathbf{v}+\mathbf{w}) = (u_x \hat{i}+ u_y\hat{j}) + (v_x \hat{i}+ v_y \hat{j}+ w_x \hat{i} + w_y \hat{j})\) Group the \(\hat{i}\) and \(\hat{j}\) components together: \(= u_x \hat{i}+ u_y\hat{j} + (v_x + w_x) \hat{i} +(v_y + w_y) \hat{j}\) Now add the components with \(\mathbf{u}\): \(= (u_x + v_x + w_x) \hat{i}+(u_y + v_y + w_y) \hat{j}\) Since we have shown that the left-hand side and the right-hand side of the equation are equal, the property is proven: $$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$$

Create a sketch to illustrate the property geometrically

To illustrate this property geometrically, we will draw the vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) and their addition. 1. Draw vector \(\mathbf{u}\) from the origin (point A) to point B. 2. Draw vector \(\mathbf{v}\) from point B to point C. This represents the sum of \(\mathbf{u}\) and \(\mathbf{v}\). 3. Draw vector \(\mathbf{w}\) from point C to point D, representing the sum of \((\mathbf{u}+\mathbf{v})+\mathbf{w}\). 4. Draw vector \(\mathbf{v}\) from point A to point E (parallelogram rule). 5. Draw vector \(\mathbf{w}\) from point E to point F, which represents the sum of \(\mathbf{v}\) and \(\mathbf{w}\). 6. Draw vector \(\mathbf{u}\) from point F to point D (parallelogram rule), representing the sum of \(\mathbf{u}+(\mathbf{v}+\mathbf{w})\). In the sketch, point D should be the same point in both the left-hand side and the right-hand side additions, illustrating that the vector addition is associative.

Key concepts

Vector Addition

Vector addition is the process of adding two or more vectors together to form a new vector. This operation is very similar to adding numbers, but it considers both magnitude and direction. When you have vectors in the plane, let's say vectors \( \mathbf{u} \) and \( \mathbf{v} \), you add them by summing their corresponding components. For example:

• If \( \mathbf{u} = u_x \hat{i} + u_y \hat{j} \) and \( \mathbf{v} = v_x \hat{i} + v_y \hat{j} \), then \( \mathbf{u} + \mathbf{v} = (u_x + v_x) \hat{i} + (u_y + v_y) \hat{j} \).

You can think of each vector as a step: \( \mathbf{u} \) moves you to a new position from the origin, and \( \mathbf{v} \) continues moving you from that position. In geometric terms, this is often illustrated by aligning the tail of the second vector, \( \mathbf{v} \), to the head of the first vector, \( \mathbf{u} \), forming a triangle or a straight path in the case of several vectors being added sequentially. This is why vector addition is useful for determining resultant forces and directions in physics.

Associative Property

The associative property is a fundamental property of addition that applies not just to scalar numbers but to vectors as well. It states that the way we group vectors when adding them does not affect the resulting vector. In mathematical terms, for three vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), associativity is defined as:

• \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\)

This means if you first add \( \mathbf{u} \) and \( \mathbf{v} \), then add \( \mathbf{w} \); or if you first add \( \mathbf{v} \) and \( \mathbf{w} \), then add \( \mathbf{u} \), you'll arrive at the same endpoint in terms of both magnitude and direction. The associative property is important because it simplifies calculations and helps in proving other vector identities. In practical applications, it means that the computation of vector sums can be rearranged and grouped in any order for convenience and efficiency without altering the result.

Component Form

The component form of a vector breaks it down into its directional elements in a coordinate system, usually the \( x \) and \( y \) axes in a 2D space. Any vector can be represented by its components along these axes, making calculations straightforward. Given a vector \( \mathbf{u} \) in component form, it is expressed as:

• \( \mathbf{u} = u_x \hat{i} + u_y \hat{j} \)

Here, \( u_x \hat{i} \) is the horizontal component, and \( u_y \hat{j} \) is the vertical component. Writing a vector this way separates its influence in each direction, which is useful for addition and subtraction.
When you add vectors like \( \mathbf{u}, \mathbf{v}, \text{ and } \mathbf{w} \) using their component forms, as in the solution above, you simply add their corresponding components. This confirms both intuitively and mathematically that vectors are additive: their components sum up directly, enhancing the aim to understand and solve problems visually and analytically.