Scalar Multiplication

Calculate c <a, b>

The vector <a,b> is multiplied by a real number c; that real number is called a scalar. To multiply a vector in component form by a scalar value, multiply each of the components by the scalar. In this example, multiply both a and b by c.

A vector has a value (magnitude) and a direction. A scalar only has a value. However, that value changes the SCALE of the vector. For example, multiplying a vector by the scalar 2 doubles its length.

c<a,b> = <ca, cb>

Multiply each of the components of vector <4,7> by 3.

Distribute –5 to each of the components of the vector.

Apply the vector magnitude formula presented in Problem 14.25.

Multiply the x- and y-components of v by the scalar 4.

In Problem 15.24, you determine that the magnitude of v is in Problem 15.25, you determine that 4v = <8,4>. Calculate the magnitude of 4v and compare it to the magnitude of v.

Vector 4v has magnitude which is 4 times the magnitude of v: This demonstrates one property of scalar multiplication: The product of a scalar and a vector’s magnitude is equal to the magnitude of the product of the scalar and the vector.

In this problem, four times the length of v is equal to the length of vector 4v.

Note that 4v = v + v + v + v. Use the head-to-tail technique demonstrated in Problems 15.1–15.11 to add those four vectors together. As the following diagram illustrates, the sum is a vector whose length must be four times the length of v, because it is literally comprised of four copies of that vector.

This problem asks you to verify that . In Problems 15.24–15.27, you verify this is true for a specific vector and a specific scalar (<2,1> and 4, respectively); in this problem you prove that the same relationship between magnitude and scalar multiplication holds true for all vectors and scalars. Apply the magnitude formula to both sides of the equation .

Factor c² out of both terms on the left side of the equation and simplify the radical expression.

Multiplying v = <a,b> by the scalar c creates a vector with length , which is c times the length of v.

Multiply the components of u by 2, multiply the components of v by 3, and add the vectors.

Multiply the components of v by 4, multiply the components of u by 1/2, and subtract the vectors.

Or you could multiply the components of u by –1/2 and add them to 4v.

Begin by calculating w.

Now calculate the magnitude of w.

According to Problem 14.38, given a vector w, the corresponding unit vector is . It is equal to the component form of the vector divided by its magnitude.

Rationalize the denominators of the components.

Multiply each of the factors by the appropriate scalar values. Note that the x-component of s is x – 1, and both of those terms must be multiplied by 4 when you compute 4s.

Alternative: Instead of multiplying r by –5 and adding 4s + (–5r), you could multiply r by 5 and subtract 5r from 4s.

When you compute –2s, make sure to distribute –2 to both terms in the x-component: –2(x – 1) = –2x + 2.

According to Problem 15.33, r – 2s = <x + 2, –2>. Apply the magnitude formula, noting that the magnitude of the vector is equal to

Square both sides of the equation to eliminate the square root symbols.

Squaring both sides could produce extra solutions that aren’t true, so you check the solutions at the end of the problem.

Factor the equation: 0 = x(x + 4). Apply the zero-product property to solve for x.

See Problems 10.12–10.13 to review the zero-product property.

There are two potential solutions: x = 0 and x = –4. Substitute each into r – 2s = <x + 2, –2> and calculate the magnitude of the resulting vectors to verify that each equals .

You conclude that when x = 0 or x = –4.

Multiply the components of w by 6, multiply the components of v by –7, and combine the vectors.

Equivalent vectors have the same component form, so begin by expressing in terms of its components.

According to Problem 15.35, 6w – 7v = <–9,24 – 4y>. Note that its x-component is already equal to the x-component of . In order for the vectors to be equivalent, the y-components must be equal as well.