91Ó°ÊÓ

The magnitude of a vector, often referred to as its length, is a measure of how long the vector is in space. It is the distance from the vector's initial point to its terminal point. For a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \), the magnitude is calculated using the formula \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \). This is essentially the three-dimensional version of the Pythagorean theorem.

In our problem, the vector \( \mathbf{v} = 5\mathbf{j} \) has a magnitude of 5 already given, because its only component is along the \( y \)-axis. For the rotating vector \( \mathbf{u} \), with formula \( 3\cos(\theta)\mathbf{i} + 3\sin(\theta)\mathbf{j} \), the magnitude remains constant at 3, since \( \cos^2(\theta) + \sin^2(\theta) = 1 \)

• A vector's magnitude provides insight into its strength without considering direction.
• Magnitudes are always non-negative values.
• For effective problem solving, know that any scalar multiply affects only magnitude, not direction.

The direction of a vector indicates the way in which it points in space. For two-dimensional vectors, direction is commonly specified by an angle with respect to a reference axis, typically the positive x-axis. When a vector is represented as \( a\mathbf{i} + b\mathbf{j} \), the angle can be found using \( \theta = \arctan\left(\frac{b}{a}\right) \).

In the exercise, vector \( \mathbf{v} = 5\mathbf{j} \) points directly along the positive y-axis, indicating it moves up vertically without any horizontal component. The vector \( \mathbf{u} \) has a direction that rotates freely in the xy-plane based on the angle \( \theta \). With each rotation, \( \mathbf{u} \) traces a circle with a radius equal to its magnitude, which is 3.

• It is essential to detach direction from magnitude when analyzing a vector's influence.
• In cross products, the resulting vector direction is determined by the respective orientations of the original vectors involved.
• The direction is vital in determining the vector's alignment and angle relative to coordinate axes.

Determinant expansion is a technique used in calculus and linear algebra to compute the cross product of two vectors. For vector cross product computation, a 3x3 matrix is formed and expanded to find a resultant vector perpendicular to the original vectors.

The matrix layout is:
\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \3\cos(\theta) & 3\sin(\theta) & 0 \0 & 5 & 0 \end{vmatrix}\]

Using cofactor expansion along the first row, the expansion leads to:
* \( (0)\mathbf{i} - (0)\mathbf{j} + (15\cos(\theta))\mathbf{k} \), resulting in the simplified vector pointing purely in the \( z \)-axis

• This method affirms that the cross product vector is orthogonal to both input vectors.
• The scalar results from the determinant can be positive, negative, or zero, directly affecting the magnitude of the cross product.
• This determinant expansion highlights the spatial relationship between vectors often used in physics to compute torques and rotational forces.