Question

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

Short answer

Question: Prove that the dot product of a vector \(\mathbf{u}\) with the sum of two vectors \(\mathbf{v}\) and \(\mathbf{w}\) is equal to the sum of the dot products of \(\mathbf{u}\) with \(\mathbf{v}\) and \(\mathbf{u}\) with \(\mathbf{w}\). Solution: We have shown that the expression \(u_{1}(v_{1}+w_{1}) + u_{2}(v_{2}+w_{2})+u_{3}(v_{3}+w_{3})\) is equal to the sum of the dot products \((u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}) + (u_{1}w_{1} + u_{2}w_{2} + u_{3}w_{3})\). Therefore, we have proven that \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}\).

Step-by-step solution

Calculate \(\mathbf{v}+\mathbf{w}\)

We begin by calculating the sum of vectors \(\mathbf{v}\) and \(\mathbf{w}\): $$\mathbf{v}+\mathbf{w} = \left\langle v_{1}+w_{1}, v_{2}+w_{2}, v_{3}+w_{3}\right\rangle$$

Calculate \(\mathbf{u} \cdot (\mathbf{v}+\mathbf{w})\)

Next, we calculate the dot product of \(\mathbf{u}\) and the sum of vectors \(\mathbf{v}\) and \(\mathbf{w}\): $$\mathbf{u} \cdot (\mathbf{v}+\mathbf{w}) = \left\langle u_{1}, u_{2}, u_{3} \right\rangle \cdot \left\langle v_{1}+w_{1}, v_{2}+w_{2}, v_{3}+w_{3} \right\rangle$$ Using the formula for dot product, this becomes: $$=u_{1}(v_{1}+w_{1}) + u_{2}(v_{2}+w_{2})+u_{3}(v_{3}+w_{3})$$

Calculate \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{u} \cdot \mathbf{w}\)

Now we calculate the dot products of vectors \(\mathbf{u}\) and \(\mathbf{v}\), and \(\mathbf{u}\) and \(\mathbf{w}\): $$\mathbf{u} \cdot \mathbf{v} = \left\langle u_{1}, u_{2}, u_{3} \right\rangle \cdot \left\langle v_{1}, v_{2}, v_{3} \right\rangle = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}$$ $$\mathbf{u} \cdot \mathbf{w} = \left\langle u_{1}, u_{2}, u_{3} \right\rangle \cdot \left\langle w_{1}, w_{2}, w_{3} \right\rangle = u_{1}w_{1} + u_{2}w_{2} + u_{3}w_{3}$$

Calculate \(\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w}\)

We then find the sum of the dot products calculated in step 3: $$\mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} = (u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}) + (u_{1}w_{1} + u_{2}w_{2} + u_{3}w_{3})$$

Compare both expressions

Finally, we compare the expressions obtained in steps 2 and 4: $$u_{1}(v_{1}+w_{1}) + u_{2}(v_{2}+w_{2})+u_{3}(v_{3}+w_{3}) = (u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}) + (u_{1}w_{1} + u_{2}w_{2} + u_{3}w_{3})$$ This equation proves the desired property: $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

Key concepts

Vector Operations

Understanding vector operations is essential for a plethora of fields, including physics, computer science, and engineering. At the core of these operations is the ability to manipulate vectors, which are entities that have both magnitude and direction.

The dot product, also known as the scalar product, is a fundamental operation in which two vectors are combined to produce a scalar value. This operation is particularly important in projecting one vector onto another and in determining the angle between vectors. In mathematical terms, if we have two vectors \(\mathbf{a}\) and \(\mathbf{b}\), their dot product is defined as \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n\), where \(a_i\) and \(b_i\) are the components of vectors \(\mathbf{a}\) and \(\mathbf{b}\), respectively.

Scalar Multiplication

Moving on to scalar multiplication, this operation involves the combination of a vector and a scalar quantity, which is a single number representing a magnitude or length. In essence, when we perform scalar multiplication, we're scaling the vector by a certain factor. As an illustration, for a vector \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and a scalar \(c\), the scalar multiplication \(c\mathbf{u}\) results in a new vector \(\langle cu_1, cu_2, cu_3 \rangle\).

This operation alters the magnitude of the vector by \(c\) times but retains its direction (unless \(c\) is negative, in which case the direction is reversed). Scalar multiplication is used in various applications such as stretching or compressing vectors, as well as in the renovation of equations in different mathematical contexts.

Vector Addition

Lastly, vector addition is the process of combining two or more vectors to form a new vector, often referred to as the resultant vector. The operation follows the principle of 'tip-to-tail' addition, where the tail of one vector is placed at the tip of another. Suppose we have vectors \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\) and \(\mathbf{w} = \langle w_1, w_2, w_3 \rangle\), their sum is found by adding the corresponding components, yielding \(\mathbf{v} + \mathbf{w} = \langle v_1 + w_1, v_2 + w_2, v_3 + w_3 \rangle\).

Vector addition is associative and commutative, which means that the order of addition does not affect the resultant vector, and it can be used to derive other vector operations, such as the distributive property showcased in the dot product exercise you might have encountered in your textbook.