Problem 10 - Algebra and Trigonometry

Question

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle 23,-5,12\rangle, \quad P(-6,4,2) $$

Step-by-Step Solution

Answer

The terminal point is (17, -1, 14).

Step 1: Understanding the Vector Components

The vector \( \mathbf{v} = \langle 23, -5, 12 \rangle \) represents the change from the initial point to the terminal point. This means that the vector tells us how much we need to move along each coordinate axis.

Step 2: Computing the Terminal Point (x-coordinate)

To find the terminal point's \(x\)-coordinate, take the initial point's \(x\)-coordinate and add the vector's \(x\) component: \(-6 + 23 = 17\).

Step 3: Computing the Terminal Point (y-coordinate)

To find the terminal point's \(y\)-coordinate, take the initial point's \(y\)-coordinate and add the vector's \(y\) component: \(4 - 5 = -1\).

Step 4: Computing the Terminal Point (z-coordinate)

To find the terminal point's \(z\)-coordinate, take the initial point's \(z\)-coordinate and add the vector's \(z\) component: \(2 + 12 = 14\).

Step 5: Final Answer

Putting these together, the terminal point is \((17, -1, 14)\).

Key Concepts

Initial Point

The initial point of a vector is where the vector starts. Think of it as the beginning of a journey, where you decide the starting position. In coordinates, the initial point is often represented by a letter with its respective coordinate values, like \(P(-6, 4, 2)\). This means our journey begins at \(-6\) in the \(x\)-axis, \(4\) in the \(y\)-axis, and \(2\) in the \(z\)-axis.

If you think of a vector as an arrow indicating direction and distance, the initial point is the tail of the arrow. It provides the starting reference needed to calculate where this journey will end, known as the terminal point.

When dealing with vectors, the initial point guides you in determining the vector’s path, helping to plot the movement accurately.

Terminal Point

The terminal point is the final destination of a vector. It's the point you reach after following the vector's direction and magnitude from the initial point.

To find the terminal point, you add the vector components to the initial point's coordinates:

• The x-coordinate is obtained by adding \(23\) (the x-component of \(\mathbf{v}\)) to the initial x of -6. This results in \(17\).
• The y-coordinate combines the initial y of \(4\) and the y-component of \(-5\) to get \(-1\).
• The z-coordinate adds \(12\) (z-component) to the initial z of \(2\) resulting in \(14\).

These calculations pinpoint the terminal point at \((17, -1, 14)\), marking the end of our vector path.

It's essential to correctly determine this ending point to understand where exactly the vector leads in space.

Vector Components

Vector components define how a vector moves from its initial point to its terminal point. They give you the changes that occur along each axis: \(x\), \(y\), and \(z\). For our vector \(\mathbf{v} = \langle 23, -5, 12 \rangle\):

• The first number, \(23\), is the x-component. It indicates how much to move in the x-direction.
• The second number, \(-5\), is the y-component, which tells you the movement in the y-direction.
• The third number, \(12\), is the z-component that specifies the movement in the z-direction.

Each component represents a vector's reach and influence along a particular axis.

The components are crucial because they help translate the vector's direction and length into tangible coordinates, allowing us to find the terminal point from a known starting location.