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When we talk about a resultant vector, we are referring to the vector that results from adding two or more vectors together. Imagine you are walking in one direction and then in another; the resultant vector is like your final position from where you started.
In the given exercise, we have vectors \( \overrightarrow{\mathbf{A}} \) and \( \overrightarrow{\mathbf{B}} \) with defined directions.
The resultant vector \( \overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} \) is what you'd get if you combine these two vectors. Its direction is given as \(60.0^\circ\) north of east.
To determine the resultant vector's direction, we consider both vectors' combined effect:

• \( \overrightarrow{\mathbf{A}} \) pushes east with a magnitude of 6.00 units.
• \( \overrightarrow{\mathbf{B}} \) pushes north, with its magnitude being solved for.

The angle \(60.0^\circ\) helps us figure out how much of the total push is going north versus east.

Magnitude calculation is crucial when determining the size or length of a vector. Think of it like measuring the distance from one point to another.
In the problem, you find the magnitude of \( \overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} \) using the Pythagorean theorem because it forms a right triangle with the components of the vectors.
Here's a quick recap of the process:

• First, resolve the vector components: \( \overrightarrow{\mathbf{A}} = (6.00, 0) \) and \( \overrightarrow{\mathbf{B}} = (0, B) \).
• Apply the theorem: \[ |\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} |= \sqrt{(6.00)^2 + (10.39)^2} \],
• Simplify to get \( |\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}| \approx 12.01 \) units.

This step ensures you know exactly how long the resulting path (vector) is when combining your movements east and north.

Understanding vector component form is essential for breaking down and analyzing vectors effectively. It’s like seeing a puzzle piece as part of a bigger picture.
Vectors are often represented by their components to simplify calculations, especially in two-dimensional space. For vector \( \overrightarrow{\mathbf{A}} \), its component form is \( (6.00, 0) \) as it only moves east. Vector \( \overrightarrow{\mathbf{B}} \), which moves north, is expressed as \( (0, B) \).
The beauty of using component form is:

• It allows you to easily add vectors together by simply adding corresponding components.
• You can better visualize vector operations, especially when considering angle and direction.

These insights make solving vector problems straightforward by separating and combining movements on the x (east-west) and y (north-south) axes.