91Ó°ÊÓ

When dealing with vector addition, unit-vector notation is an efficient and precise way to express vectors. This method utilizes specific symbols for directions in three-dimensional space. The most common unit vectors are:

• \( \hat{\mathrm{i}} \) for the x-axis direction
• \( \hat{\mathrm{j}} \) for the y-axis direction
• \( \hat{\mathrm{k}} \) for the z-axis direction

Using unit-vector notation, we can succinctly write vectors as combinations of these unit vectors with their respective components. Take the vectors \( \vec{a} \) and \( \vec{b} \) given in the exercise:

• \( \vec{a} = (4.0 \mathrm{\ m}) \hat{\mathrm{i}} + (3.0 \mathrm{\ m}) \hat{\mathrm{k}} \)
• \( \vec{b} = (-13.0 \mathrm{\ m}) \hat{\mathrm{i}} + (7.0 \mathrm{\ m}) \hat{\mathrm{k}} \)

Notice how each component is paired with a unit vector, indicating its direction along the axes. To add two vectors given in this form, simply add their corresponding components. This step is fundamental for combining vectors effectively and forms the basis for more complex vector manipulations.

The magnitude of a vector represents its length or size and is an essential aspect of vector analysis. It provides information about how "long" a vector is in its given space, independent of its direction. For a vector \( \vec{v} = (v_x \hat{\mathrm{i}} + v_y \hat{\mathrm{k}}) \), the magnitude is calculated using the Pythagorean theorem as follows: \[ \|\vec{v}\| = \sqrt{v_x^2 + v_y^2} \]Applying this formula to the sum vector \( \vec{a} + \vec{b} = (-9.0 \mathrm{\ m}) \hat{\mathrm{i}} + (10.0 \mathrm{\ m}) \hat{\mathrm{k}} \), we find: \[\| \vec{a} + \vec{b} \| = \sqrt{(-9.0)^2 + (10.0)^2} = \sqrt{181} \approx 13.45 \mathrm{\ m}\]The magnitude is a key feature that helps in understanding the overall effect of vector quantities in physics and engineering. It's crucial particularly in cases where force, velocity, or any other vector quantity is analyzed for its impact.

The direction of a vector is equally important as its magnitude because it tells us where the vector is headed in space. For two-dimensional vectors lying in the \( \hat{\mathrm{i}} \)-\( \hat{\mathrm{k}} \) plane, the direction can be found using the arctangent function: \[\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)\] Direction is measured as an angle, often in degrees. For the vector sum \( \vec{a} + \vec{b} \), we calculate:\[\theta = \tan^{-1}\left(\frac{10.0}{-9.0}\right) \approx \tan^{-1}(-1.11) \approx -48.01^\circ\]This angle is measured from the negative \( \hat{\mathrm{i}} \) axis toward the positive \( \hat{\mathrm{k}} \) direction. It is crucial to understand the role of direction to fully grasp vectors' influence in a given context. Remember, angles can be adjusted contextually, often needing interpretation as to where they are measured from.