Evaluate the given expression with 𝐮=(-2,-1,4,5), 𝐯=(3,1,-5,7), and 𝐰=(-6,2,1,1). (a) 3 𝐮-5 𝐯+𝐰

step 1 Hello guys. So for this exercise we got these three vectors here. So for the part A we need to c

Evaluate the given expression with 𝐮=(-2,-1,4,5),...

Key Concepts

Vector Addition and Subtraction

In vector algebra, vector addition involves combining two or more vectors by adding their corresponding components. This process is fundamental in producing resultant vectors from linear combinations, where vectors are added or subtracted component-wise. The operation is essential in solving problems involving displacement, forces, or any vector quantities, without specific reliance on their numerical values in any given example.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a real number (scalar), which scales each component of the vector by that number. This operation retains the direction of the vector (unless the scalar is negative, which reverses its direction) and is a building block for forming linear combinations, as well as for expressing relationships and transformations in various applications of linear algebra.

Euclidean Norm of a Vector

The Euclidean norm, often simply called the magnitude or length of a vector, is a measure of the distance of the vector from the origin in Euclidean space. It is computed by taking the square root of the sum of the squares of its components. This measure is critical in assessing the size of vectors and is widely used in fields ranging from physics to data science to quantify distances and errors.

Norm Properties: Homogeneity and Triangle Inequality

The norm function exhibits important properties such as homogeneity and the triangle inequality. Homogeneity means that scaling a vector by a real number scales its norm by the absolute value of that number. The triangle inequality signifies that the norm of the sum of two vectors is less than or equal to the sum of their individual norms. These properties facilitate the manipulation and simplification of expressions involving vector norms in a broad array of mathematical and applied contexts.

Transcript

0:00 Hello guys.

00:01 So for this exercise we got these three vectors here.

00:07 So for the part a we need to compute the norm of three times the vector u minus five times the vector v and plus the vector w.

00:23 Okay, so we need to compute this norm.

00:26 So let's first compute the summation inside of the norm.

00:31 So three u is just minus 6, minus 3, and 12, 15.

00:42 Okay.

00:44 Then minus 5 times the vector v is minus 15, minus 5, 25, and minus 35.

01:04 Okay.

01:05 And w, well, w is just the same vector, is minus 6, 2, 2.

01:11 Two, one, one.

01:13 So let's sum all these three vectors together and we obtain that, so 3u minus 5b plus w is 27 minus 6, minus 6, 38 and minus 19.

01:50 Okay, and now let's compute the norm.

01:53 So the norm of 3u minus 5b plus w is, well the square root of 27 square plus 6 square plus 38 square plus 19 square.

02:13 And this is the square root of 200, of 200, 250, 2 ,000, 5 ,000, 570.

02:30 6 -70 that's the solution now let's see what happened with the part b which is the norm of 3 times the vector u minus 5 times the norm of the vector v and plus the norm of w so we know that we can take this out and we just have 3 the norm of u and this is just 3 times the square root of 2 square plus 1 plus 4 square plus 5 square and this is just 3 times the square root of 46 then we got minus 5 the vector v the normal the vector b so this is minus 5 the square root here of of 3 squared plus 1 plus 5 squared plus 7 squared...

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