What is the angle in radians between the vectors?

Take the dot product of the two vectors
a ˙ b = -7*(-10) + 6*7 + (-3)*(-2) = 118

Now you know that
a ˙ b = |a||b| cos(θ)
So we also need to calculate |a| and |b|
|a| = sqrt[7^2+6^2+3^2] = 9.69
|b| = sqrt(10^2+7^2+2^2) = 12.37

therefore we have
9.69*12.37 * cos(θ) = 118
cos(θ) = 118/(9.69*12.37) = .9844
take the arccos to get θ = .1766 radians.

Notice that since the cos(θ) is close to 1 you know that the angle is close to 0.

Given vectors X and Y. X = 2i - j + 2k Y = 5i + j - 3k (a) discover the attitude ?, in radians between them. Calculate the magnitue of the vectors. we can use those later. || X || = ?[2² + (-a million)² + 2²) = ?(4 + a million + 4) = ?9 = 3 || Y || = ?[5² + a million² + (-3)²] = ?(25 + a million + 9) = ?35 Now calculate the dot made from X and Y. X • Y = <2, -a million, 2> • <5, a million, -3> = 10 - a million - 6 = 3 The dot product is often expressed in a distinct way. X • Y = || X || || Y || cos? 3 = (3?35)cos? cos? = a million/?35 ? = arccos(a million/?35) ? a million.40095 radians ________ (b) stumble on a vector of length a million it extremely is perpendicular to both X and Y. First take the go product. n = X x Y = <2, -a million, 2> x <5, a million, -3> = <a million, 16, 7> Calculate the importance of vector n. || n || = ?(a million² + 16² + 7²) = ?(a million + 256 + 40 9) = ?306 Divide by utilising the importance of n to get a unit vector perpendicular to both vectors X and Y. n/?306 = <a million/?306, 16/?306, 7/?306>

a • b = IaI IbI cos Ө

(118)=[49+36+9]^(1/2)[100+49+4]^(1/2)cosӨ

(118) = [94]^(1/2) [ 153]^(1/2) cos Ө

cos Ө = 118 / [ 94^(1/2) 153^(1/2) ]

cos Ө = 0.984

Ө = 10.3 º