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you have a 3 by 3 chessboard with 3 white pawns and 3 black pawns. given white goes first and both teams play their best what is the outcome? is the outcome a draw always since pawns can only move forward and kill diagonally. This means that the middle pawn can only kill either the left or...

Let H be a subset of R3, defined by: W={(x, y, z) : x - y + 1 = e^z}. Is H a subspace of R3? To determine if H is a subspace of R3, we need to satisfy the three properties. The zero vector of R3 is in H because substituting x = y = z = 0 into x - y + 1 = e^0 gives us a valid expression. I am...

The right angle triangle has legs of length r and a right angle at O. A quarter of a circle is consturcted with the centre O and a radius of r. A semicircle is construted with a diameter of AB. Prove that the area of the shaded lune is equal to the area of triangle AOB. Area of the...

Prove that if 4 is subtracted from the square of an integer greater than 3, the result is a composite number. I have let k represent any positve integer. therefore k + 3 > 3. (k+3)^2 - 4 k^2 + 6k + 9 - 4 k^2 + 6k + 5 I am stuck here and don't know how to proceed. If somebody could help that...

We have been given 8 cases and several systems of equations to solve using matrices rref. After solving the matrices we need to state which case it is. The cases only include three planes. So if I have two equations with three variables to solve for, would this be classfied as one of the cases?

The problem is: Show that the line of intersection of the planes 2x + y - 3z = 1 and -2x + 3y + 2z = 4 is perpendicular to the line r = (1, 3, 2) + t(2, -3, -2). I have worked my way through this problem and I have the parametric equations for the line of intersection. I am stuck here and...

If you have two vectors and they are perpendicular, if you use the dot product on them they will add to zero. How can you determine if they are parallel?

Word problem: A plane is headed N 40° E with a speed of 600 A plane is headed N 40° E with a speed of 600 km/h. A west wind causes it to travel N 45° E. Find the resultant speed of the plane and the speed of the wind. I am having difficulties starting this question, if somebody could give me...