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Homework Statement
explain why the vectors a=(-2,3) and b=(3,-1) = (13,9)
how do i do this
A vector is a mathematical object that has both magnitude (size) and direction. It can be represented by an arrow, with the length of the arrow indicating the magnitude and the direction of the arrow indicating the direction.
The magnitude of a vector can be calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In the case of a vector, the magnitude is equal to the square root of the sum of the squares of the components of the vector. In this case, the magnitude of vector a=(-2,3) is equal to √((-2)^2+(3)^2) = √(4+9) = √13. Similarly, the magnitude of vector b=(3,-1) is equal to √(3^2+(-1)^2) = √(9+1) = √10.
The direction of a vector is determined by the angle it makes with a reference axis. This angle can be calculated using trigonometric functions such as tangent or sine. In this case, the direction of vector a=(-2,3) can be calculated as tan⁻¹(3/-2) ≈ -56.31 degrees. Similarly, the direction of vector b=(3,-1) can be calculated as tan⁻¹(-1/3) ≈ -18.43 degrees.
To add two vectors, you must first align them so that their tails are at the same point. Then, you can add their components to get the resulting vector. In this case, the sum of vectors a=(-2,3) and b=(3,-1) will be (a+b)=(a₁+b₁, a₂+b₂) = (-2+3, 3+(-1)) = (1, 2).
A vector can be represented in component form by listing its components in order, separated by a comma. In this case, the vectors a and b can be represented as a=(-2,3) and b=(3,-1).