Question

In Exercises 77 and \(78,\) use vectors to find the point that lies two-thirds of the way from \(P\) to \(Q\). \(P(4,3,0), \quad Q(1,-3,3)\)

Short answer

The point that lies two-thirds of the way from P to Q has coordinates (2, -1, 2).

Step-by-step solution

Finding the PQ Vector

Subtract the coordinate values of P from Q to get the PQ vector. The vector PQ can be described as Q - P, giving us (1-4, -3-3, 3-0) = (-3, -6, 3).

Calculate Two-thirds of PQ Vector

To find the point that lies two-thirds of the way from P to Q, we need to find two-thirds of the PQ vector. We do this by multiplying each component of the PQ vector by 2/3, resulting in (-3*2/3, -6*2/3, 3*2/3) = (-2, -4, 2).

Adding to P

Finally, add the calculated two-third vector to the P coordinate. The resulting coordinates are (4-2, 3-4, 0+2) = (2, -1, 2).

Key concepts

Vectors in Calculus

Vectors are fundamental in calculus for representing quantities that have both magnitude and direction. They are denoted as an array of numbers, called components, which correspond to coordinates in space. In the context of vector calculus, operations such as addition, subtraction, and scalar multiplication are essential to solving problems related to motion, force, and fields in physics and engineering.

Diving deeper, vector calculus uses vectors to describe phenomena in multi-dimensional space. For instance, in the exercise provided, we are dealing with three-dimensional vectors, where each component represents the vector's extension in the x, y, and z directions respectively. By understanding vectors, students can tackle a range of problems from simple point-to-point movements to complex physical systems described by vector fields.

Finding Points Using Vectors

Finding a specific point using vectors involves a few key vector operations. In the given exercise, the goal is to locate a point that is a certain fraction of the way from point P to point Q. This is a common task in vector calculus, which can be visualized as a point moving along the line segment connecting P and Q.

To find this point, first, the vector representing the line segment, or the displacement vector, must be calculated. This is done by subtracting the coordinates of the starting point P from the ending point Q. Once the displacement vector is known, multiplying it by the desired fraction gives a new vector that reaches from P towards Q. Finally, adding this vector to the coordinates of point P places it correctly on the line segment. This method of finding intermediate points is invaluable in fields such as computer graphics and navigation.

Vector Operations

Vector operations are the tools we use to manipulate and combine vectors. The most basic are addition, subtraction, and scalar multiplication. In our exercise, scalar multiplication is used when we multiply the vector PQ by two-thirds, scaling its length without changing its direction. The resulting vector represents two-thirds of the journey from P to Q.

Subtraction was also used at the beginning to find the vector PQ. It's important to note that vector addition and subtraction are done component-wise. This means we handle each dimension independently, which is why we subtract coordinate values of P from Q individually. Moreover, scalar multiplication also works component-wise, as each component of the vector is multiplied by the scalar.

These operations are essential for translating, rotating, and scaling objects in space, and serve as a foundation for more complex operations like the dot product and cross product, which in turn are fundamental in applications such as work calculation and determining orthogonality or rotational direction, respectively.