91Ó°ÊÓ

The scalar product, also known as the dot product, is a fundamental operation in vector mathematics. When we calculate the dot product of two vectors \( \vec{A} \) and \( \vec{B} \), it results in a single scalar value. This operation is defined as:

• \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \)

Here, \( |\vec{A}| \) and \( |\vec{B}| \) are the magnitudes (lengths) of the vectors, and \( \theta \) is the angle between them. The term \( \cos(\theta) \) helps determine how aligned the vectors are.
When the scalar product is positive, the vectors point roughly in the same direction. Conversely, when it's negative, as in our exercise, the vectors face more towards opposite directions. A zero result indicates perpendicular vectors. In the given exercise, the scalar product is \(-6.00\), implying a significant component of one vector opposes the other.

The vector product or cross product results in another vector rather than a scalar. It measures how two vectors "cross" each other. The magnitude of the vector product is given by:

• \( |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin(\theta) \)

In this formula, \( \sin(\theta) \) plays a key role in determining how perpendicular the vectors are to each other.
The cross product is maximal when the vectors are perpendicular, as \( \sin(90^\circ) = 1 \).
In our context, the vector product has a magnitude of \(+9.00\), suggesting a substantial perpendicular component. The direction of the resultant vector from a cross product can be determined using the right-hand rule, which is crucial in applications such as physics and engineering.

Trigonometric identities are equations involving trigonometrical functions that are true for all values of the involved variables. These identities are essential for solving problems involving angles and dimensions, like the one in this exercise.

• One key identity used here is \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

Using this relationship, we can solve for the angle \( \theta \) between vectors when both \( \sin(\theta) \) and \( \cos(\theta) \) are known.
In our exercise, since \( \tan(\theta) = \frac{9.00}{-6.00} = -1.5 \), this indicates a negative tangent, which occurs in either the second or fourth quadrant. Given our conditions, the fourth quadrant is appropriate.
Understanding these identities helps not just in vector mathematics, but across various mathematical and scientific disciplines.