Given that m=6i+8j and n=-8i+7/3j
Magnitude and direction of m and n
Angles between m and n
To find the magnitude of a vector, we can use the formula:
|m| = √(x² + y²)
For m = 6i + 8j, we have:
|m| = √(6² + 8²)
= √(36 + 64)
= √100
= 10
Therefore, the magnitude of vector m is 10.
To find the magnitude of n = -8i + (7/3)j, we have:
|n| = √((-8)² + (7/3)²)
= √(64 + 49/9)
= √(576/9 + 49/9)
= √(625/9)
= 25/3
Therefore, the magnitude of vector n is 25/3.
To find the direction of a vector, we can use the formula:
θ = tan^(-1)(y/x)
For m = 6i + 8j, we have:
θ_m = tan^(-1)(8/6)
= tan^(-1)(4/3)
≈ 53.13°
Therefore, the direction of vector m is approximately 53.13°.
For n = -8i + (7/3)j, we have:
θ_n = tan^(-1)((7/3)/(-8))
= tan^(-1)(-7/24)
≈ -16.26°
Therefore, the direction of vector n is approximately -16.26°.
To find the angle between m and n, we can use the formula:
θ = cos^(-1)((m * n) / (|m| * |n|))
Where (m * n) is the dot product of vectors m and n. The dot product is given by:
m * n = (m_x * n_x) + (m_y * n_y)
For m = 6i + 8j and n = -8i + (7/3)j, we have:
(m * n) = (6 * -8) + (8 * (7/3))
= -48 + (56/3)
= -144/3 + 56/3
= -88/3
Plugging in the values, we get:
θ = cos^(-1)((-88/3) / (10 * (25/3)))
≈ 172°
Therefore, the angle between vectors m and n is approximately 172°.
To find the magnitude of a vector, you can use the formula: magnitude = √(x² + y²)
1. Magnitude of vector m:
m = 6i + 8j
Magnitude of m = √(6² + 8²)
= √(36 + 64)
= √100
= 10
2. Magnitude of vector n:
n = -8i + (7/3)j
Magnitude of n = √((-8)² + (7/3)²)
= √(64 + 49/9)
= √(576/9 + 49/9)
= √(625/9)
= 25/3
To find the direction of a vector, you can use the formula: direction = atan2(y, x)
3. Direction of vector m:
m = 6i + 8j
Direction of m = atan2(8, 6)
= arctan(8/6)
≈ 53.13°
4. Direction of vector n:
n = -8i + (7/3)j
Direction of n = atan2(7/3, -8)
= arctan((7/3)/-8)
≈ -111.80°
To find the angle between two vectors, you can use the formula: angle = arccos((m⋅n) / (|m|⋅|n|))
where "." denotes the dot product between two vectors.
5. Angle between vectors m and n:
m = 6i + 8j
n = -8i + (7/3)j
First, calculate the dot product:
m⋅n = (6*(-8)) + (8*(7/3))
= -48 + 56/3
= -144/3 + 56/3
= -88/3
Then, calculate the magnitudes:
|m| = 10
|n| = 25/3
Finally, calculate the angle:
angle = arccos((-88/3) / (10 * (25/3)))
= arccos((-88/3) / (250/3))
= arccos(-88/250)
≈ 98.45°