Verify Equation (4.7 .6) for the given vector. 𝐯=(-1,2,0) ; V, B, and C from Problem 20.

Name: Problem 34, Chapter 4: Differential Equations and Linear Algebra
Uploaded: 2020-05-22T18:18:07-08:00
Duration: 9 min 59 s
Channel: Arsalan Afkhami
Description: Verify Equation (4.7 .6) for the given vector. 𝐯=(-1,2,0) ; V, B, and C from Problem 20.

Verify Equation (4.7 .6) for the given vector. 𝐯=(-1,2,0) ;...

Key Concepts

Verification of Vector Equations

Verifying a vector equation involves substituting known vector values into both sides of the equation and showing that they yield the same vector. This process requires careful computation using the rules of vector operations and is a key skill in linear algebra and vector analysis.

3-Dimensional Euclidean Space

This concept involves the understanding that vectors exist in a space defined by three dimensions, where each vector is represented by an ordered triple. It is crucial for working with spatial problems, as every vector has components that can be manipulated through addition, scaling, and other operations in this space.

Vector Operations

Vector operations include addition, scalar multiplication, and often the dot or cross product. These operations form the foundation for manipulating and combining vectors, allowing us to simplify and verify expressions involving vectors by performing computations component-wise.

Linear Combination of Vectors

A linear combination refers to expressing a vector as the sum of scalar multiples of other vectors. This concept is fundamental in understanding how vectors can be decomposed or represented in terms of a chosen set of basis vectors or other related vectors, which is often essential in verifying vector identities.

Transcript

00:01 For the given vector, first we need to find the vector relative to the given basis b and c.

00:13 So to do that, i write the vector which is negative 1 to 0.

00:21 And then we need to find a vector c1, c2, c3.

00:29 That's when multiplied by the ordered basis b gives us the vector so what i mean is that if i write c1 multiplied by negative 7 4 which is the same vector of ordered basis b plus c2 4 2 c3 maybe 750 here we have x y z and x and x is c1 multiplied by negative 7 of the x of the first vector c2 multiplied by the x of the second vector and c3 multiplied by the x of the third vector so we have a system of equations negative 7 multiplied c1 and i continue to write the rest.

01:42 It's equal to x.

01:47 1 and 4c1 plus 2c2 plus 5c3.

02:01 It is equal to 2 .4c1 minus c2 is equal to 0.

02:11 You just need to simplify it so c1 is equal to c2 divided by 4 then you replace c1 here and then rewrite as some equations with two equations so 7 divided 4 c2 plus 4 c2 minus 7 c3 so equals to negative 1.

02:52 And then here we have c2 plus 2, c2, plus 5, c3, is supposed to 2.

03:03 Just we can simplify here the c2.

03:06 And then we have two equations that we can simply solve.

03:11 And then we get c1 and c2, which will give us the vector relative to the order basis b equals to a c3.

03:31 So the first vector, the vector relative to order basis b is 3 divided by 43, 12 divided by 43, and 10 divided by 43.

03:48 Now we also need to find the vector relative to the order basis c.

03:56 So it's similar to what i wrote already...

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