Question

On a diagram show the arbitrary vectors \(p\) and q. Then show the following: (a) \(p+q\) (d) \(4 q\) (e) \(-2 q\) (c) \(q-p\).

Short answer

Question: Represent the following vector operations for arbitrary vectors p and q: (a) p + q, (b) 4q, (c) -2q, and (d) q - p. Answer: (a) To represent p + q, place the initial point of vector q at the end point of vector p and draw a vector from the initial point of p to the end point of q. (b) To represent 4q, multiply the magnitude of q by 4 while keeping the direction unchanged. (c) To represent -2q, multiply the magnitude of q by 2 but reverse its direction. (d) To represent q - p, first find -p by rotating vector p by 180 degrees, then place the initial point of vector -p at the end point of vector q, and draw a vector from the initial point of q to the end point of -p.

Step-by-step solution

Vector Addition (p + q)

To find the p + q, place the initial point of vector q at the end point of vector p. Now, draw a vector from the initial point of vector p to the end point of vector q. This new vector represents the sum p + q.

Multiplication by Scalar (4q)

To multiply vector q by a scalar, we must change the magnitude of the vector q while keeping its direction unchanged. In this case, the scalar is 4, so the new vector (4q) will have 4 times the length of the original vector q in the same direction as q.

Multiplication by Scalar (-2q)

To multiply vector q by a scalar, we must change the magnitude of the vector q while keeping its direction unchanged. In this case, the scalar is -2, so the new vector (-2q) will have 2 times the length of the original vector q but in the opposite direction as q because of the negative sign.

Vector Subtraction (q - p)

To subtract p from q, we start by finding -p. To find -p, rotate vector p by 180 degrees. Now, we can treat q - p as an addition of vectors q and -p. Hence, place the initial point of vector -p at the end point of vector q, and then draw a vector from the initial point of vector q to the end point of vector -p. This new vector represents the difference q - p.

Key concepts

Understanding Vector Addition

Vector addition is a core concept in understanding how vectors interact with each other. When we talk about the addition of two vectors, say vectors \( p \) and \( q \), we think of them as quantities that can be added using geometric methods.
To perform vector addition, place the tail (initial point) of vector \( q \) at the head (end point) of vector \( p \). This way, you create a continuous line leading from the start of vector \( p \) to the end of vector \( q \). The resultant vector, \( p + q \), is the vector that starts at the initial point of vector \( p \) and ends at the final point of vector \( q \).

Here are some important notes about vector addition:

• It’s commutative: \( p + q = q + p \).
• It’s associative: \((p + q) + r = p + (q + r) \).
• The zero vector \( \mathbf{0} \) is the additive identity: \( p + \mathbf{0} = p \).

Understanding these properties helps simplify multi-step vector problems and solves complex tasks easily.

Scalar Multiplication of Vectors

Scalar multiplication involves adjusting the magnitude of a vector without altering its direction. When multiplying a vector \( q \) by a scalar, such as 4 or -2, you change the vector's length in a specific manner.

Consider the scalar multiplication of a vector \( q \) by 4. This transformation scales the vector, making it four times longer than the original while pointing in the same direction. On the contrary, multiplying by a negative scalar, like -2, flips the direction of the vector 180 degrees while doubling its length.

The key points to remember about scalar multiplication include:

• The direction of the vector remains the same unless multiplied by a negative number.
• A positive scalar lengthens the vector, whereas a fractional scalar shortens it.
• A zero scalar results in the zero vector, effectively nullifying the original vector's properties.

By mastering these details, solving problems involving scaling vectors becomes much clearer.

Insights into Vector Subtraction

Vector subtraction might initially seem confusing, yet it becomes straightforward by borrowing ideas from vector addition. Subtraction of vectors, such as \( q - p \), involves adding the negative of a vector.
To perform vector subtraction, you first find the negative of vector \( p \), which mathematically flips the direction of \( p \). This transformation is equivalent to multiplying \( p \) by -1. Then, you add the resulting vector (\(-p\)) to vector \( q \) using vector addition techniques.

Key observations on vector subtraction are:

• Subtraction is not commutative: \( q - p eq p - q \).
• The negative vector \(-p\) has the same magnitude as \( p \) but in the opposite direction.
• The result of \( q - p \) is the vector from the tip of \( p \) directly to the tip of \( q \).

Simplifying vector subtraction by thinking in terms of vector addition helps comprehend the underpinning mechanics of vectors.