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. $$(a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}$$

Short answer

Question: Prove the vector property $$(a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}$$ using components and provide a brief geometric illustration of the property.

Step-by-step solution

Represent \(\mathbf{v}\) in component form

Write the vector \(\mathbf{v}\) as an ordered pair of its components in the \(x y\)-plane, denoted with \(v_x\) and \(v_y\). The vector will be represented as follows: $$\mathbf{v} = \langle v_x, v_y \rangle$$

Perform the scalar multiplications

Perform the scalar multiplication of \((a+c)\) on \(\mathbf{v}\) and the separate scalar multiplications of \(a\) and \(c\) on \(\mathbf{v}\): $$(a+c) \mathbf{v} = (a+c) \langle v_x, v_y \rangle = \langle (a+c)v_x, (a+c)v_y \rangle$$ $$a \mathbf{v} = a\langle v_x, v_y \rangle = \langle av_x, av_y \rangle$$ $$c \mathbf{v} = c\langle v_x, v_y \rangle = \langle cv_x, cv_y \rangle$$

Add the separate scalar multiplications

Add the two separate scalar multiplications, \(a \mathbf{v}\) and \(c \mathbf{v}\), component-wise: $$a\mathbf{v} + c\mathbf{v} = \langle av_x, av_y \rangle + \langle cv_x, cv_y \rangle = \langle av_x + cv_x, av_y + cv_y \rangle$$

Prove the property

Show that the result from the scalar multiplication of \((a+c)\) on \(\mathbf{v}\) is equal to the sum of the separate scalar multiplications of \(a\) and \(c\) on \(\mathbf{v}\): $$\langle (a+c)v_x, (a+c)v_y \rangle = \langle av_x + cv_x, av_y + cv_y \rangle$$ Since the components of both sides are equal, we have proven the property: $$(a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}$$

Geometrical illustration

To visualize this property geometrically, the sketch will include three vectors: \((a+c) \mathbf{v}\), \(a \mathbf{v}\), and \(c \mathbf{v}\). The sum \(a \mathbf{v} + c \mathbf{v}\) will also be depicted, showing that it is equal to \((a+c) \mathbf{v}\). 1. Draw the vector \(\mathbf{v}\) in the \(x y\)-plane with its tail at the origin. 2. Draw the vector \((a+c) \mathbf{v}\), which is the arrow from the origin to the point on the line of \(\mathbf{v}\) with length \((a+c)\) times the length of \(\mathbf{v}\). 3. Draw the vectors \(a \mathbf{v}\) and \(c \mathbf{v}\), which are arrows from the origin to the points on the line of \(\mathbf{v}\) with lengths \(a\) and \(c\) times the length of \(\mathbf{v}\), respectively. 4. Place the tail of the vector \(c \mathbf{v}\) at the tip of the vector \(a \mathbf{v}\) to form the vector sum \(a \mathbf{v} + c \mathbf{v}\). Observe that the tip of \(c \mathbf{v}\) lies on the same point as the tip of \((a+c) \mathbf{v}\). This sketch shows geometrically that the property \((a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}\) is valid.

Key concepts

Component Form

The component form of a vector is a way of expressing a vector using its individual parts or coordinates. In a 2-dimensional plane, this is usually written as an ordered pair, representing the vector's horizontal and vertical components. For a vector \( \mathbf{v} \) in the \( xy \)-plane, it can be represented as \( \langle v_x, v_y \rangle \), where \( v_x \) is the component along the x-axis, and \( v_y \) is the component along the y-axis.

• Describing Component Form: This form is particularly useful because it allows for easy calculation and manipulation of vectors.
• Visualization: You can think of each component as describing movement along a particular direction: \( v_x \) moves right or left, and \( v_y \) moves up or down.
• Advantages: Makes mathematical operations like addition and scalar multiplication straightforward since they can be applied component-wise.

Understanding component form simplifies many vector operations such as addition and scalar multiplication, which we will explore further.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a number, known as a scalar. This operation affects each component of the vector by stretching or compressing it. Let's consider the vector \( \mathbf{v} = \langle v_x, v_y \rangle \). If you multiply \( \mathbf{v} \) by a scalar \( a \), it becomes \( a \mathbf{v} = \langle av_x, av_y \rangle \).

• Effects of Scalar Multiplication:
  • If \( a > 1 \), the vector stretches longer.
  • If \( 0 < a < 1 \), the vector shrinks.
  • If \( a < 0 \), the vector flips direction.
• Real-world Analogy: Think of a vector as an arrow on paper. Scalar multiplication is like resizing this arrow without changing its direction unless the scalar is negative.

Scalar multiplication is essential when proving vector properties such as \((a+c) \mathbf{v}=a \mathbf{v}+c \mathbf{v}\), as it allows us to scale vectors to different desired lengths.

Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. In component form, you simply add the corresponding components of the vectors. For two vectors, \( \mathbf{u} = \langle u_x, u_y \rangle \) and \( \mathbf{v} = \langle v_x, v_y \rangle \), their sum is \( \mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle \).

• Understanding Vector Addition:
  • This reflects how vectors can be lined up, with the tail of one meeting the head of the other.
  • The resulting vector, or resultant, spans the beginning of the first vector to the end of the last.
• Geometric Interpretation: Often depicted using the "tip-to-tail" method where vectors are placed sequentially to find a resultant.

Vector addition is key in solving problems like the given property \((a+c) \mathbf{v} = a \mathbf{v} + c \mathbf{v}\), demonstrating that scalar products can be added to yield the original scalar applied to the sum.