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°