- Identify the components of the vector by calculating the difference between the terminal and initial point coordinates: \(a = 8 - 5 = 3\) and \(b = 4 - 6 = -2\).
- Represent the vector in component form: \(\vec{v} = (3, -2)\).
- Express the vector as a linear combination of the unit vectors \(\vec{i}\) and \(\vec{j}\): \(\vec{v} = 3\vec{i} - 2\vec{j}\).
- Match the result with the given options, identifying the correct expression as \(\boxed{3 \vec{i} - 2 \vec{j}}\).
Explanation
- Understanding the Problem and Definitions
We are given an initial point \(P_1 = (5, 6)\) and a terminal point \(P_2 = (8, 4)\) for a vector \(\vec{v}\). A vector in the Cartesian coordinate system is defined by the difference between the terminal point coordinates and the initial point coordinates. If a vector has an initial point \((x_1, y_1)\) and a terminal point \((x_2, y_2)\), its components \((a, b)\) are given by \(a = x_2 - x_1\) and \(b = y_2 - y_1\). We are also asked to express the resulting vector as a linear combination of the standard unit vectors \(\vec{i} = (1, 0)\) and \(\vec{j} = (0, 1)\). A vector \((a, b)\) can be written as \(a\vec{i} + b\vec{j}\).
- Calculating Vector Components
To find the components \((a, b)\) of vector \(\vec{v}\), we subtract the coordinates of the initial point from those of the terminal point:
\[a = x_2 - x_1 = 8 - 5 = 3\]
\[b = y_2 - y_1 = 4 - 6 = -2\]
Thus, the vector \(\vec{v}\) in component form is \((3, -2)\).
- Finalizing the Vector Expression
Now that we have the components of the vector, we express it as a linear combination of the standard unit vectors \(\vec{i}\) and \(\vec{j}\). Using the definition of vector components, we have:
\[\vec{v} = (3, -2) = 3(1, 0) + (-2)(0, 1) = 3\vec{i} - 2\vec{j}\]
Comparing this result to the provided options:
○ \(\vec{v} = -2 \vec{i} + 3 \vec{j}\)
○ \(\vec{v} = -3 \vec{i} - 2 \vec{j}\)
○ \(\vec{v} = -3 \vec{i} + 2 \vec{j}\)
○ \(\vec{v} = 3 \vec{i} - 2 \vec{j}\)
○ \(\vec{v} = 3 \vec{i} + 2 \vec{j}\)
○ None of the above.
The vector matches the fourth option. Therefore, the vector expressed as a linear combination of unit vectors is \(\boxed{3 \vec{i} - 2 \vec{j}}\).
Examples
In navigation and aviation, pilots use vectors to describe movement between two points on a map. If a pilot starts at a location \((x_1, y_1)\) and travels to a destination \((x_2, y_2)\), the displacement vector represents the change in position. Understanding how to express this vector as a linear combination of unit vectors helps in calculating the ground speed and direction, allowing the pilot to adjust their course for wind or other environmental factors efficiently. Knowing the horizontal and vertical components of the flight path ensures that the pilot reaches their destination accurately.