Question #c976d

Question

Question #c976d

1 Answer

Impact of this question

2702 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-26T16:35:07

See below

Explanation:

Given two vectors $\to a, \to b$ $vec a, vec b$ their sum is

$\to s = \to a + \to b$ $vec s = vec a + vec b$ . Now computing the norm of $\to s$ $vec s$ and squaring

${∥ ∥ \to s ∥ ∥}^{2} = {∥ ∥ ∥ \to a + \to b ∥ ∥ ∥}^{2} = {∥ ∥ \to a ∥ ∥}^{2} + 2 ⟨ \to a, \to b ⟩ + {∥ ∥ ∥ \to b ∥ ∥ ∥}^{2}$ $norm (vec s)^2 = norm( vec a + vec b)^2 = norm (vec a)^2+2 << vec a, vec b >> + norm( vec b)^2$

The scalar product

$⟨ \to a, \to b ⟩ = ∥ ∥ \to a ∥ ∥ ∥ ∥ ∥ \to b ∥ ∥ ∥ cos (ˆ \to a, \to b)$ $<< vec a, vec b >> = norm vec a norm vec b cos(hat(vec a, vec b))$ has a maximum when $cos (ˆ \to a, \to b) = 1$ $cos(hat(vec a, vec b))=1$ and a minimum when $cos (ˆ \to a, \to b) = - 1$ $cos(hat(vec a, vec b))=-1$ so

$min norm(vec s)^2 = norm (vec a)^2+norm(vec b)^2-2norm vec a norm vec b$ and

$max norm(vec s)^2 = norm (vec a)^2+norm(vec b)^2+2norm vec a norm vec b$

Finally, when two vectors $vec a, vec b$ are aligned, their sum is a minimum or a maximum deppending on their relative orientation:

concordant a maximum
discordant a minimum

Science

Math

Humanities

... and beyond

Question #c976d

1 Answer

Explanation:

Related questions

Impact of this question