Given the vectors a = (1, 3, 4) and b = (4, 5, -4), what is the angle between the two given vectors to the nearest degree?
Calculate the cross product using the matrix determinant rule. That will tell you the sine of the angle.
The magnitudes of the two vectors are
sqrt(1+9+16) = 5.099 and sqrt(16 + 25 + 16) = 7.550
The cross product is
|i j k |
|1 3 4 |
|4 5 -4 |
= -24 i + 20 j -7 k
The magnitude of the cross product is
sqrt (576 + 400 + 49) = 32.015
sin theta = 32.015 /[(5.099)(7.550)]
= 0.8316
theta = 56.3 degrees
wow, thanks so much :) it really helped.
To find the angle between two vectors, you can use the dot product formula. The dot product of two vectors a and b is given by the formula:
a · b = |a| * |b| * cosθ
where |a| and |b| represent the magnitudes (or lengths) of vectors a and b, and θ is the angle between them.
In this case, we have a = (1, 3, 4) and b = (4, 5, -4).
Step 1: Find the magnitudes of vectors a and b.
The magnitude of a vector v = (x, y, z) is given by the formula:
|v| = √(x^2 + y^2 + z^2)
For vector a, we have:
|a| = √(1^2 + 3^2 + 4^2)
= √(1 + 9 + 16)
= √26
For vector b, we have:
|b| = √(4^2 + 5^2 + (-4)^2)
= √(16 + 25 + 16)
= √57
Step 2: Calculate the dot product of vectors a and b.
The dot product of vectors a and b is given by the formula:
a · b = (1 * 4) + (3 * 5) + (4 * -4)
= 4 + 15 - 16
= 3
Step 3: Find the angle θ using the dot product formula.
Using the dot product formula:
a · b = |a| * |b| * cosθ
3 = (√26) * (√57) * cosθ
Divide both sides by (√26) * (√57):
cosθ = 3 / (√26) * (√57)
Calculate the value of cosθ and find the angle θ using the inverse cosine function (cos^-1).
θ = cos^-1(3 / (√26) * (√57))
Using a calculator, we find that θ is approximately 66 degrees.
Therefore, the angle between the given vectors to the nearest degree is 66 degrees.