Find the norm of a linear combination of vectors, given their norms

This question already has an answer here:

  • Vectors and Norms

    4 answers

Solutions Collecting From Web of "Find the norm of a linear combination of vectors, given their norms"

First compute A=angle between a and b using law of cosines $$\Vert a+b\Vert^2=\Vert a\Vert^2 +\Vert b\Vert^2 + 2\Vert a\Vert \Vert b\Vert cos(a,b)$$
7×7 =4×4 + 5×5 + 2x4x5 cosA ,
cosA=1/5 ,

Norm of $$\Vert a-b\Vert^2=\Vert a\Vert^2 +\Vert b\Vert^2 – 2\Vert(a)\Vert b\Vert cos(a,b)$$
$$\Vert(a-b)\Vert^2 =16 +25 -2.4.5.1/5=33$$
Checking $$ 4 \Vert a\Vert \Vert b\Vert=\Vert a+b\Vert^2 -\Vert a-b\Vert^2$$
4x4x5x1/5=16=49-33=16 , it’s OK

Again use law of cosines $$\Vert 2a +(-3b)\Vert^2 =\Vert2a\Vert^2 + \Vert 3b\Vert^2 -2\Vert 2a\Vert.\Vert3b\Vert cos A$$
$$\Vert 2a +(-3b)\Vert^2=4.16 + 9.25 -2.8.15.1/5 $$
$$\Vert 2a -3b \Vert^2 =241$$ , $$\Vert (2a -3b)\Vert=15.52$$