Adding and Subtracting Vectors Algebraically

If \(\mathbf{v}=\langle 2,3,-1\rangle \) and \(\mathbf{w}=\langle -1,-2, 4\rangle \), find:

1. \(\mathbf{v}+\mathbf{w}\)
2. \(\mathbf{w-v}\)
3. \(\dfrac{1}{2}\mathbf{w}\)
4. \(2\mathbf{v}+3\mathbf{w}\)

Solution

(a) \(\mathbf{v} + \mathbf{w} = \langle 2,3,-1\rangle +\langle -1,-2,4\rangle =\left\langle 2+(-1), 3+(-2), -1+4\right\rangle =\langle 1,1,3\rangle\)

(b) \(\mathbf{w-v=} \langle -1,-2,4\rangle - \langle 2,3,-1\rangle =\left\langle -1-2, -2-3, 4-( -1) \right\rangle =\left\langle -3,-5,5\right\rangle \)

(c) \(\dfrac{1}{2}\mathbf{w}=\dfrac{1}{2}\,\langle -1,-2,4\rangle = \left\langle \dfrac{1}{2}( -1), \dfrac{1}{2} ( -2), \dfrac{1}{2}(4) \right\rangle = \left\langle -\dfrac{1}{2},-1,2\right\rangle \)

(d) \(2\mathbf{v} + 3\mathbf{w} = 2\,\langle 2,3,-1\rangle+3\,\langle -1,-2,4\rangle =\langle 4,6,-2\rangle +\langle -3,-6,12\rangle =\langle 1,0,10\rangle\)