If vector OA=4i+2j ,OB=6i-8j and OC=-2i+5j then find the value of <ACB.
To find the value of ∠ACB, we can use the dot product formula for finding the angle between two vectors.
The dot product of two vectors A and B is given by the formula:
A · B = |A| * |B| * cos(θ)
Where A · B represents the dot product, |A| and |B| represent the magnitudes of vectors A and B, and θ represents the angle between the two vectors.
In this case, we can consider vectors OA, OB, and OC as position vectors (vectors starting from the origin) of points A, B, and C, respectively.
Now, we can start solving this problem step by step:
1. Calculate the magnitudes of vectors OA, OB, and OC.
|OA| = √(4^2 + 2^2) = √(16 + 4) = √20 = 2√5
|OB| = √(6^2 + (-8)^2) = √(36 + 64) = √100 = 10
|OC| = √((-2)^2 + 5^2) = √(4 + 25) = √29
2. Calculate the dot product of vectors OA and OB.
OA · OB = (4)(6) + (2)(-8) = 24 - 16 = 8
3. Calculate the dot product of vectors OC and OB.
OC · OB = (-2)(6) + (5)(-8) = -12 - 40 = -52
4. Calculate the dot product of vectors OC and OA.
OC · OA = (-2)(4) + (5)(2) = -8 + 10 = 2
5. Calculate the cosine value of the angle ∠ACB using the dot product formula:
cos(∠ACB) = (OC · OB) / (|OC| * |OB|) => -52 / (√29 * 10)
6. Use the inverse cosine (arccos) function to find the value of ∠ACB:
∠ACB = arccos(cos(∠ACB))
So, the value of ∠ACB is equal to arccos(-52 / (√29 * 10)) in radians or degrees, depending on the units you are using.