Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = <5, 4> , b = <2, −1>

To find the angle between two vectors, we can use the dot product formula:

a · b = |a| |b| cos(θ)

where a · b is the dot product of vectors a and b, |a| is the magnitude of vector a, |b| is the magnitude of vector b, and θ is the angle between them.

First, let's calculate the magnitudes of vectors a and b:

|a| = √(5² + 4²) = √(25 + 16) = √41
|b| = √(2² + (-1)²) = √(4 + 1) = √5

Now, let's calculate the dot product of vectors a and b:

a · b = (5)(2) + (4)(-1) = 10 - 4 = 6

Substituting the values into the formula, we have:

6 = √41 √5 cos(θ)

To find cos(θ), divide both sides of the equation by √41 √5:

cos(θ) = 6 / (√41 √5)

Now, find the value of cos(θ) using a calculator:

cos(θ) ≈ 0.4364

To find θ, we can take the inverse cosine (arccos) of 0.4364:

θ ≈ arccos(0.4364)

Using a calculator, we find:

θ ≈ 63.8 degrees (approximated to the nearest degree)

Therefore, the angle between the vectors a and b is approximately 63.8 degrees.

To find the angle between two vectors, we can use the dot product formula:

a · b = |a| |b| cos(θ)

Where a · b denotes the dot product of vectors a and b, |a| and |b| represent the magnitude (or length) of a and b respectively, and θ is the angle between them.

First, let's calculate the dot product of vectors a and b:

a · b = (5)(2) + (4)(-1) = 10 - 4 = 6

Next, we need to calculate the magnitudes of vectors a and b:

|a| = sqrt(5^2 + 4^2) = sqrt(25 + 16) = sqrt(41)

|b| = sqrt(2^2 + (-1)^2) = sqrt(4 + 1) = sqrt(5)

Now, we can substitute these values into the dot product formula and solve for the angle θ:

6 = sqrt(41) sqrt(5) cos(θ)

Divide both sides by sqrt(41) sqrt(5) to isolate cos(θ):

cos(θ) = 6 / (sqrt(41) sqrt(5))

Using a calculator, we can evaluate this expression to get an approximate value of cos(θ):

cos(θ) ≈ 0.673

To find θ, we need to find the value of cos^(-1)(0.673), which represents the inverse cosine function. On most calculators, this is denoted by "cos^(-1)" or "arccos".

Using a calculator, we find:

θ ≈ cos^(-1)(0.673) ≈ 47.76° (rounded to two decimal places)

Therefore, the exact expression for the angle between vectors a and b is cos^(-1)(0.673), and when approximated to the nearest degree, it is approximately 48°.

come on, basic stuff

a dot b = |a| |b|cosØ, where Ø is the angle between them
<5, 4>dot<2, −1> = √(25+16)√(4+1)cosØ
10 - 4 = √205cosØ
cosØ = 6/√205
Ø = appr 65.2°