Parallel and orthogonal

I need math help in this problem: a=(-5, 5 3) b=(-2,-4,-5) (they are vectors)
Decompose the vector b into b=v+w where v is parallel to a and w is orthogonal to a, find v and w

Correct result:

v₁ =  2.1186
v₂ =  0
v₃ =  0
w₁ =  0
w₂ =  0
w₃ =  0

Solution:

$$\begin{gathered} a=(-5,5,3) \\ b=(-2,-4,-5) \\ b=v+w \\ v\parallel a=>v=ka \\ w\perp a=>w\mathrm{.}a=0 \\ -2=v_1+w_1 \\ -4=v_2+w_2 \\ -5=v_3+w_3 \\ {v}_1=-5\cdot k=2.1186 \\ {v}_2=5\cdot k \\ {v}_3=3\cdot k \\ {w}_1\cdot (-5)+w_2\cdot 5+w_3\cdot 3=0 \\ {v}_1+w_1=-2 \\ {v}_2+w_2=-4 \\ {v}_3+w_3=-5 \\ 5k+v_1=0 \\ 5k-v_2=0 \\ 3k-v_3=0 \\ 5w_1-5w_2-3w_3=0 \\ k={\displaystyle \frac{-25}{59}}\doteq -0.423729 \\ {v}_1={\displaystyle \frac{125}{59}}\doteq 2.118644 \\ {v}_2={\displaystyle \frac{-125}{59}}\doteq -2.118644 \\ {v}_3={\displaystyle \frac{-75}{59}}\doteq -1.271186 \\ {w}_1={\displaystyle \frac{-243}{59}}\doteq -4.118644 \\ {w}_2={\displaystyle \frac{-111}{59}}\doteq -1.881356 \\ {w}_3={\displaystyle \frac{-220}{59}}\doteq -3.728814 \end{gathered}$$

$v_{2}=0$

$v_{3}=0$

$w_{1}=0$

$w_{2}=0$

$w_{3}=0$

Try another example

Showing 0 comments:

Next similar math problems:

• Here is
  Here is a data set (n=117) that has been sorted. 10.4 12.2 14.3 15.3 17.1 17.8 18 18.6 19.1 19.9 19.9 20.3 20.6 20.7 20.7 21.2 21.3 22 22.1 22.3 22.8 23 23 23.1 23.5 24.1 24.1 24.4 24.5 24.8 24.9 25.4 25.4 25.5 25.7 25.9 26 26.1 26.2 26.7 26.8 27.5 27.6 2
• Speed of Slovakian trains
  Rudolf decided to take the train from the station 'Ostratice' to 'Horné Ozorovce'. In the train timetables found train Os 5409 : km 0 Chynorany 15:17 5 Ostratice 15:23 15:23 8 Rybany 15:27 15:27 10 Dolné Naštice 15:31 15:31 14 Bánovce nad Bebravou 15:35 1
• Three points 2
  The three points A(3, 8), B(6, 2) and C(10, 2). The point D is such that the line DA is perpendicular to AB and DC is parallel to AB. Calculate the coordinates of D.
• Vector perpendicular
  Find the vector a = (2, y, z) so that a⊥ b and a ⊥ c where b = (-1, 4, 2) and c = (3, -3, -1)
• The raw
  The raw data presented here are the scores (out of 100 marks) of a market survey regarding the acceptability of new product launched by a company for random sample of 50 respondents: 40 45 41 45 45 30 30 8 48 25 26 9 23 24 26 29 8 40 41 42 39 35 18 25 35
• Eq2 2
  Solve following equation with quadratic members and rational function: (x²+1)/(x-4) + (x²-1)/(x+3) = 23
• Vector v4
  Find the vector v4 perpendicular to vectors v1 = (1, 1, 1, -1), v2 = (1, 1, -1, 1) and v3 = (0, 0, 1, 1)
• Three parallels
  The vertices of an equilateral triangle lie on 3 different parallel lines. The middle line is 5 m and 3 m distant from the end lines. Calculate the height of this triangle.
• Vector equation
  Let’s v = (1, 2, 1), u = (0, -1, 3) and w = (1, 0, 7) . Solve the vector equation c1 v + c2 u + c3 w = 0 for variables c1 c2, c3 and decide weather v, u and w are linear dependent or independent
• Z9–I–4 MO 2017
  Numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 were prepared for a train journey with three wagons. They wanted to sit out so that three numbers were seated in each carriage and the largest of each of the three was equal to the sum of the remaining two. The conduct
• Candles
  Before Christmas, Eva bought two cylindrical candles - red and green. Red was 1 cm longer than green. She lit a red candle on Christmas Day at 5:30 p. M. , lit a green candle at 7:00 p. M. , and left them both on fire until they burned. At 9:30 p. M. , bo
• Four prisms
  Question No. 1: The prism has the dimensions a = 2.5 cm, b = 100 mm, c = 12 cm. What is its volume? a) 3000 cm² b) 300 cm² c) 3000 cm³ d) 300 cm³ Question No.2: The base of the prism is a rhombus with a side length of 30 cm and a height of 27 cm. The heig
• Pool
  If water flows into the pool by two inlets, fill the whole for 19 hours. The first inlet filled pool 5 hour longer than the second. How long pool take to fill with two inlets separately?
• Two trains
  There are two trains running the same distance. 1st train will travel it in 7 hours 21minutes. 2nd the train will travel 5 hours 57minutes and 34 seconds and it is 14 km/h faster than the first train. What are speeds of trains and how long is this railway
• Two planes
  Two planes flying from airports A and B, 420 km distant against each other. Plane from A took off 15 minutes later and flies at an average speed of 40 km/h higher than the plane from B. Determine the average speed of this two aircraft if you know that it
• Cuboids
  Two separate cuboids with different orientation in space. Determine the angle between them, knowing the direction cosine matrix for each separate cuboid. u1=(0.62955056, 0.094432584, 0.77119944) u2=(0.14484653, 0.9208101, 0.36211633)
• Motel
  Three different dinners A, B, C are served in the motel. Three 20 members went to motel. Group 1 ordered dinners: A 20 members and their average dinner price in the group was CZK 200. From Group 2, they ordered a A: 10 persons, B 10 dinners, and their ave