Question

Find the following vectors. The vector that is 3 times (3,-5) plus -9 times (6,0)

Short answer

Question: Calculate the vector that is 3 times the first vector (3,-5) plus -9 times the second vector (6,0). Answer: The resulting vector is (-45, -15).

Step-by-step solution

Scalar multiplication of each vector

First, we need to perform scalar multiplication on each vector. Scalar multiplication means multiplying a vector by a scalar (a number). To find 3 times (3,-5), multiply each component of the vector by 3: 3 * (3, -5) = (3*3, 3*(-5)) = (9, -15) To find -9 times (6,0), multiply each component of the vector by -9: -9 * (6, 0) = (-9*6, -9*0) = (-54, 0) Now we have two new vectors: (9, -15) and (-54, 0).

Vector addition

Now that we have the scalar multiplication of each vector, we can add these two vectors together to get the final result. To add two vectors, we simply add their corresponding components: (9, -15) + (-54, 0) = (9 + (-54), -15 + 0) = (-45, -15) So, the vector that is 3 times (3,-5) plus -9 times (6,0) is (-45, -15).

Key concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector arithmetic that involves multiplying each component of a vector by a constant, known as a scalar. In simpler terms, if you have a vector represented by \( (x, y) \) in two-dimensional space, and a scalar represented by a real number \( a \) , then the scalar multiplication of the vector is performed as \( a(x, y) = (a\times x, a\times y) \) .

Let's take an example. If we have a scalar value of 3 and a vector \( (3, -5) \) , the scalar multiplication process will yield \( (3\times3, 3\times(-5)) = (9, -15) \) . This operation stretches or shrinks the vector based on the scalar's value and can even reverse its direction if the scalar is negative. Scalar multiplication is essential as it allows us to resize vectors while keeping their direction intact, which is a key aspect of many vector operations.

This concept is applied directly in the given exercise, where a vector is multiplied by a positive and a negative scalar, demonstrating both stretching and direction reversal.

Vector Addition

Vector addition is an operation that combines two or more vectors into a single resultant vector by adding their corresponding components. When adding vectors in two-dimensional space (often denoted as \( R^2 \) ), simply add the x-components together, and then the y-components.

The summed vector's components are thus calculated by \( (x_1, y_1) + (x_2, y_2) = (x_1+x_2, y_1+y_2) \) . Here's an example: if you have vectors \( (9, -15) \) and \( (-54, 0) \) , the resultant vector will be \( (9 + (-54), -15 + 0) = (-45, -15) \) .

Vector addition is commutative, meaning the order of the vectors added does not affect the result. Whether you add \( A \) to \( B \) or \( B \) to \( A \) , the resultant vector will be the same. This property simplifies many calculations as it allows for flexibility in the order of operations. The exercise demonstrates vector addition as the final step after scalar multiplication has been performed on the individual vectors.

Linear Combinations of Vectors

A linear combination of vectors involves creating a new vector by adding together multiple vectors each multiplied by their respective scalar coefficients. Essentially, it’s combining scalar multiplication and vector addition. The general form of a linear combination for two vectors in \( R^2 \) is \( a(v_1) + b(v_2) \) where \( v_1 \) and \( v_2 \) are vectors and \( a \) and \( b \) are scalars.

For instance, if we have two vectors \( (3, -5) \) and \( (6, 0) \) and two scalars 3 and -9, the linear combination would be \( 3(3, -5) + -9(6, 0) \) . When you perform these operations, you’re essentially creating a new vector that is a “blend” of the original vectors, scaled by the given amounts. This process is crucial in many areas of mathematics and physics as it allows for the construction of new vectors that possess qualities derived from the base vectors. The exercise presented is an application of linear combinations of vectors.

Vectors in R2

Vectors in \( R^2 \) refer to vectors existing in a two-dimensional space. This plane is characterized by having two axes, typically the x-axis and the y-axis. In this space, any vector can be represented as an ordered pair \( (x, y) \) describing its horizontal (x-component) and vertical (y-component) displacement. The concept of vectors in \( R^2 \) is vital for visualizing and solving many problems in physics, engineering, and mathematics.

Vectors in two dimensions can be manipulated through various operations, such as scalar multiplication, vector addition, and finding linear combinations as shown in the exercise. They provide a way to concisely represent and compute direction and magnitude for quantities like velocity and force. Understanding vectors in \( R^2 \) lays the foundation for more advanced topics in linear algebra and vector calculus, where these concepts are extended to three dimensions or more.