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The cross product of two vectors is a unique operation that results in a new vector that is perpendicular to the plane formed by the original vectors. Specifically, for two-dimensional vectors, such as the ones given here, \(\vec{a} = 3.0 \hat{\mathrm{i}} + 5.0 \hat{\mathrm{j}}\) and \(\vec{b} = 2.0 \hat{\mathrm{i}} + 4.0 \hat{\mathrm{j}}\), the cross product produces a vector along the z-axis (denoted by \(\hat{k}\)).

The formula for the cross product in 2D is:

• \(\vec{a} \times \vec{b} = (a_1b_2 - a_2b_1) \hat{k}\)

By substituting the values: \(3.0 \cdot 4.0 - 5.0 \cdot 2.0 = 12.0 - 10.0 = 2.0\).

Therefore, the result of the cross product here is \(2 \hat{k}\). This operation is particularly important in physics where understanding torque or rotational forces is necessary.

The dot product is a fundamental operation that takes two vectors and returns a scalar. This operation measures the extent to which two vectors point in the same direction. For the vectors given, the calculation is straightforward:

To find the dot product, use this formula:

• \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2\)

Substituting in our values might look like: \(3.0 \cdot 2.0 + 5.0 \cdot 4.0 = 6.0 + 20.0 = 26.0\).

Thus, the dot product of \(\vec{a}\) and \(\vec{b}\) equals \(26.0\). This result emphasizes the extent of alignment between these vectors, helping in applications like calculating work done by a force.

Adding vectors involves adding corresponding components from each vector to create a new vector. This process essentially combines the effects or directions of two vectors.

Here's how you'd approach it:

• For \(\vec{a} + \vec{b}\): Add the i-components and j-components separately

Calculate: \(3.0 + 2.0 = 5.0 \, and \, 5.0 + 4.0 = 9.0\). The resultant vector is \(5.0\hat{i} + 9.0\hat{j}\).

Next, compute the dot product of this sum with \(\vec{b}\):
\( (5.0 \cdot 2.0) + (9.0 \cdot 4.0) = 10.0 + 36.0 = 46.0\).
This operation is common in physics and engineering scenarios where multiple forces act simultaneously on an object.

Understanding vector components helps us break down a vector into parts that lie along the axes or along another vector. To find the component of \(\vec{a}\) along \(\vec{b}\), imagine how much of \(\vec{a}\) is going in the direction of \(\vec{b}\).

Use the following formula for the projection of \(\vec{a}\) onto \(\vec{b}\):

• \( \text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\|\vec{b}\|} \)

First, the magnitude \(\|\vec{b}\| = \sqrt{2.0^2 + 4.0^2} = \sqrt{4.0 + 16.0} = \sqrt{20.0} = 2\sqrt{5.0}\). Thus, \(\text{comp}_{\vec{b}}\vec{a} = \frac{26.0}{2\sqrt{5.0}} = \frac{13.0}{\sqrt{5.0}}\), approximately \(5.8\).

Breaking down vectors into their components is used in various fields, from physics to computer graphics, to simplify complex vector equations and problems.