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Linear Algebra
Dot Product and Orthogonal
Answer the following with complete solutions:
1.) Given the following vectors: u=(1,2,-4) and v=(2, 3, 5)
Find the angle between u and v and their unit vector/s
2.) Suppose that B=(2, 2, 1) and makes 30 degrees with A, and A*B=6. What is the magnitude of A?
3.) What is the angle between A and B if A*B=0?
4.) Find (u+v)*(2u-v) given that u*u=4, u*v=5 and v*v=10.
5.) Determine if u and v are orthogonal if u=(cos(theta), sin(theta), -1); v=(sin(theta), -cos(theta), 0).
Answer the following with complete solutions:
1.) Given the following vectors: u=(1,2,-4) and v=(2, 3, 5)
Find the angle between u and v and their unit vector/s
2.) Suppose that B=(2, 2, 1) and makes 30 degrees with A, and A*B=6. What is the magnitude of A?
3.) What is the angle between A and B if A*B=0?
4.) Find (u+v)*(2u-v) given that u*u=4, u*v=5 and v*v=10.
5.) Determine if u and v are orthogonal if u=(cos(theta), sin(theta), -1); v=(sin(theta), -cos(theta), 0).
Linear Algebra
Orthogonal
Is it true that if x is orthogonal to v and w, then x is also orthogonal to v − w. Why so?
Is it true that if x is orthogonal to v and w, then x is also orthogonal to v − w. Why so?
Linear Algebra
Dot Product
Suppose that u and v are vectors in R to the seventh power both of length 2 the square root of 2 and that the length of u − v is also2 the square root of 2 . Then parallel to u plus v parallel-to = ____________ and the angle between u and v is _____________.
Suppose that u and v are vectors in R to the seventh power both of length 2 the square root of 2 and that the length of u − v is also2 the square root of 2 . Then parallel to u plus v parallel-to = ____________ and the angle between u and v is _____________.
Linear Algebra
Let v be a vector space over F, define a spanning of v.
Linear Algebra
Write the vector v=( 1, -2, 5) as a linear combination of the vector e1=( 1, 1, 1), e2= ( 1, 2, 3) and e3=( 2, -1, 1)
Linear Algebra
show that ( 1, 1, 1), ( 0, 1, 1) and ( 0, 1, -1) generate or spanned R^3
Linear Algebra
Let V be the subspace of R3 spanned by
{(1, 1, 0), (1, 1, 1)} and T : V —> V be defined by T(x1, x2, x3) = (0, x1, x2).
Find the kernel of T.
Linear Algebra
If V is a finite dimensional vector space and v not equal to 0 is a vector in V, show that there is a linear functional f E V* such that f(v) not equal to 0
Linear Algebra
Solve by Gaussian elimination method the
following system of equations :
x+y+z+t=5
x-y+z+t=1
x + z + t = 3
following system of equations :
x+y+z+t=5
x-y+z+t=1
x + z + t = 3
Linear Algebra
Find the radius and the centre of the circular section of the sphere I r I = 4 cut off by the plane r . (2i -j + 4k) = 3.
