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The dot product is a vector operation that takes two vectors and returns a scalar. When you calculate the dot product of two vectors, you are essentially measuring how much one vector extends in the direction of another.
Imagine two vectors, **\( \mathbf{A} \)** and **\( \mathbf{B} \)**. The dot product is given by the formula:

• **\( \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \)**,

where **\( a_1, a_2, a_3 \)** are components of **\( \mathbf{A} \)** and **\( b_1, b_2, b_3 \)** are components of **\( \mathbf{B} \)**.
This formula helps find the sum of the products of the corresponding components of the two vectors.
In the example given, \overline{OA} = (2 \mathbf{i} + 3 \mathbf{j} - \mathbf{k}) and \overline{OB} = (\mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k}). When computed, the dot product is **\(-7\)**. This value indicates the vectors are not parallel or perpendicular, as neither results in zero.

Unlike the dot product, the cross product of two vectors results in a new vector that is perpendicular to the plane containing the original vectors. This operation is especially useful in physics and engineering to find a direction that is orthogonal to two given directions.
The cross product is calculated using the determinant of a matrix composed of the unit vectors and components of the two vectors. If \( \overline{OA} \) and \( \overline{OB} \) are given by the vectors **\( (2 \mathbf{i} + 3 \mathbf{j} - \mathbf{k}) \)** and **\( (\mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k}) \)** respectively, the cross product is given as:

• \[ \overline{OA} \times \overline{OB} = \left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 3 & -1 \ 1 & -2 & 3 \end{array}\right| \]

Solving this yields the vector **\(-5 \mathbf{i} - 9 \mathbf{j} + 9 \mathbf{k} \)**.
This resulting vector points orthogonally to both \( \overline{OA} \) and \( \overline{OB} \), capturing the essence of rotational direction.

The cosine of the angle between two vectors is a measure of how parallel the vectors are. Calculated using the dot product, it serves as a vital metric in various applications, from physical simulations to computer graphics.
To find the cosine of the angle \( \theta \) between vectors **\( \overline{OA} \)** and **\( \overline{OB} \)**, the formula is:

• \[ \cos \theta = \frac{\overline{OA} \cdot \overline{OB}}{\|\overline{OA}\| \|\overline{OB}\|} \].

Here, **\( \|\overline{OA}\| \)** and **\( \|\overline{OB}\| \)** represent the magnitudes of **\( \overline{OA} \)** and **\( \overline{OB} \)** respectively.
In this scenario, both magnitudes were calculated as **\( \sqrt{14} \)**, and the dot product as **\(-7\)**, leading to:

• \( \cos \theta = \frac{-7}{\sqrt{14} \cdot \sqrt{14}} = -\frac{1}{2}\).

A cosine value of **\(-\frac{1}{2}\)** corresponds to an angle of 120 degrees, signaling that the vectors form an obtuse angle.