+0

# Vectors

0
397
2

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.

v = i - j, w = i + 2j

Guest Mar 18, 2017
Sort:

#1
+76946
0

v =  < 1, - 1 >

w = < 1, 2  >

v1   = [ v (dot) w  ] / l w l^2  *  < w >

v(dot) w   =   1* 1 + 2* -1   =  1 - 2    =   -1

l w l  =  sqrt (1^2 + 5^2)   =  sqrt (5)           l w l ^2   = 5

So......

v1   =   -1 /  (5)  *  <1,  2 >     =   < -1/5, -2/ 5  >

v2  =  v - v1  =  < 1 - -1/5, -1 - - 2/5 >  =  < 1 + 1/5, -1 + 2/5 > =   < 6/5,-3/5 >

Check

Sum of v1 and v2    =  < -1/5 + 6/5, -2/5 + - 3/5 >   =  < 1  -1 >    = v

CPhill  Mar 19, 2017
#2
+18612
0

Decompose v into two vectors v1 and v2,

where v1 is parallel to w and v2 is orthogonal to w.

v = i - j, w = i + 2j

$$\vec{v} = \binom{1}{-1}\\ \vec{w} = \binom{1}{2}$$

$$\begin{array}{rcll} \vec{v_1} &=& \lambda \cdot \vec{w} \\ \vec{v_2} &=& \mu \cdot \vec{w_\perp} \\ \hline \vec{v}=\vec{v_1}+\vec{v_2} &=& \lambda \cdot \vec{w} + \mu \cdot \vec{w_\perp} \\ \vec{v} &=& \lambda \cdot \vec{w} + \mu \cdot \vec{w_\perp} \quad &| \quad \cdot \vec{w} \\ \vec{v}\cdot \vec{w} &=& \lambda \cdot \vec{w}\cdot \vec{w} + \mu \cdot \vec{w_\perp} \cdot \vec{w} \quad &| \quad \vec{w_\perp} \cdot \vec{w} = 0 \quad \vec{w}\cdot \vec{w} = w^2 = 1^2+2^2 = 5\\ \vec{v}\cdot \vec{w} &=& \lambda \cdot w^2 +0 \\ \vec{v}\cdot \vec{w} &=& \lambda \cdot w^2 \quad &| \quad : w^2 \\ \lambda &=& \frac{ \vec{v}\cdot \vec{w} } {w^2} \\ \mathbf{ \vec{v_1} } &\mathbf{=}&\mathbf { \left( \frac{ \vec{v}\cdot \vec{w} } {w^2} \right) \cdot \vec{w} } \\ \mathbf{ \vec{v_2} } &\mathbf{=}&\mathbf { \vec{v} - \vec{v1} }\\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{ \vec{v_1} } &\mathbf{=}&\mathbf { \left( \frac{ \vec{v}\cdot \vec{w} } {w^2} \right) \cdot \vec{w} } \\ & = & \left( \frac{ \binom{1}{-1}\cdot \binom{1}{2} } {5} \right) \cdot \binom{1}{2} \\ & = & \left( \frac{ 1-2 } {5} \right) \cdot \binom{1}{2} \\ & = & \left( \frac{ -1 } {5} \right) \cdot \binom{1}{2} \\ \mathbf{ \vec{v_1} } &\mathbf{=}&\mathbf { \binom{-0.2}{-0.4} } \\\\ \mathbf{ \vec{v_2} } &\mathbf{=}&\mathbf { \binom{1}{-1} - \vec{v1} }\\ & = & \binom{1}{-1} - \binom{-0.2}{-0.4} \\ & = & \binom{1}{-1} + \binom{0.2}{0.4} \\ & = & \binom{1+0.2}{-1+0.4} \\ \mathbf{ \vec{v_2} } &\mathbf{=}&\mathbf { \binom{1.2}{-0.6} } \\ \hline \end{array}$$

heureka  Mar 20, 2017

### 16 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details