Geometry and Trigonometry

Furthermore

Official Guidance, clarification and syllabus links:

Applications of the properties of the scalar product
v·w=w·v;
u·(v+w)=u·v+u·w;
(kv)·w=k(v·w);
v·v=|v|².
v·w=|v||w|cos θ, where θ is the angle between v and w.
For non-zero vectors v·w=0 is equivalent to the vectors being perpendicular;
for parallel vectors |v·w|=|v||w|.

Formula Booklet:

Scalar product	\(\mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3, \quad \text{where } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} , \text{ and }\mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix} \) \(\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}|| \mathbf{w}| \cos{\theta}, \text{ where } \theta \) is the angle between \(\mathbf{v} \) and \( \mathbf{w}\)
Angle between two vectors	\( \cos{\theta} = \dfrac{v_1w_1 + v_2w_2 + v_3w_3}{|\mathbf{v}|| \mathbf{w}|} \)

The scalar product of two vectors, also known as the dot product, is a mathematical operation that takes two vectors and returns a scalar quantity. The scalar product is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them.

The formula for the scalar product of two vectors, a and b, is:

$$ \mathbf{a \cdot b} = |\mathbf{a}| |\mathbf{b}| \cos \theta $$

where theta \( \theta \) is the angle between the vectors a and b.

A second way to calculate the scalar product is as follow:

The alternate method of finding the scalar product involves multiplying together the corresponding components of the vectors. Let's say we have two vectors a = \( a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and b = \( b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \), where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors in the \(x, y,\) and \(z\) directions, respectively. Then the scalar product of a and b is given by:

$$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$$

This can also be written in a more compact form using summation notation:

$$\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^3 a_ib_i$$

The dot product can be used find the angle between two vectors.

As an example, let's find the angle between the following two vectors:

$$ \mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ -1 \\ 2 \end{pmatrix} $$

Rearranging the formula above:

$$ \cos \theta = \frac{\mathbf{a \cdot b}}{|\mathbf{a}| |\mathbf{b}|} = \frac{(2 \times 5) + (3 \times -1) + (4 \times 2)}{\sqrt{2^2 + 3^2 + 4^2} \sqrt{5^2 + (-1)^2 + 2^2}} = 0.5085... $$ $$ \theta \approx 59.4^o $$

Using the absolute value of the scalar product when finding the angle between two vectors ensures that the acute angle is found.

Applications of the properties of the scalar product:

• Commutativity: v\(\cdot\) w = w \(\cdot\) v

• Distributivity: u \(\cdot\) (v + w) = u \(\cdot\) v + u \(\cdot\) w

• Associativity with scalar multiplication: (kv) \(\cdot\) w = k(v \(\cdot\) w)

• Self-product: v \(\cdot\) v = \(|\textbf{v}|^2\)

• Geometric interpretation: v \(\cdot\) w = \(|\textbf{v}| |\textbf{w}| \cos \theta \), where \( \theta \) is the angle between v and w

• Perpendicularity: If v \(\cdot\) w = 0, then v and w are perpendicular

• Parallelism: If v and w are parallel, then \(|\textbf{v} \cdot \textbf{w}| = |\textbf{v}| |\textbf{w}| \)