Vector algebra. Given two vectors such that a+b= 11 x̂-ŷ+5 ẑ and a-b=-5 𝐱̂+11 ŷ+9 ẑ (a) Find a a

step 1 In this question we have two vectors. A plus A and B, we are given the sum and difference of the

Vector algebra. Given two vectors such that a+b= 11 x̂-ŷ+5 ẑ...

Key Concepts

Vector Addition and Subtraction

This concept refers to the process of combining and subtracting vectors using their components. Typically, vector addition involves adding corresponding components, while subtraction involves subtracting corresponding components, which is fundamental in physics and engineering problems to compute resultant quantities or to isolate individual vectors from known combinations.

Solving Systems of Vector Equations

When provided with relationships such as the sum and difference of two vectors, one can use techniques analogous to solving algebraic equations. This involves operations like adding or subtracting the given expressions to isolate each vector. Such methods leverage linear combinations to find individual components of unknown vectors, making it an essential strategy in vector algebra.

Dot Product and Angle Between Vectors

The dot product is a scalar operation on two vectors that reflects the product of their magnitudes and the cosine of the angle between them. It is central to finding the angle between two vectors since dividing the dot product by the product of the vectors’ magnitudes yields the cosine of the included angle. This method is widely used in physics and mathematics to analyze directional relationships.

Transcript

00:01 In this question we have two vectors.

00:03 A plus a and b, we are given the sum and difference of these two vectors.

00:08 We have to find a vector a and b in the first part.

00:12 As opposed a plus b is 11x minus y plus 5g and a minus b minus b and a minus b plus x plus 11 by plus 9.

00:20 If we add these two, if we add these two vectors but again we can find here and we this plus a and minus oh sorry plus b and minus b will cancel out.

00:30 You will get 2a here which will be equal to 11156x plus 10y plus 14c and from this dividing it by 2 you will get vector a here and vector a will be equals to 3x plus 5y plus 7g this is the vector a now putting vector a in any of these we will get factor b if i have a plus b, if i put in a plus b, we can find vector b as, if i write here, b will be equals to, i have vector a, 11 is the a plus b, 11 minus.

01:27 I can write it as 11 minus 3x minus 1, minus 5, plus plus minus 1, minus 5, plus minus 1, minus 1, minus 5 it will be y and 5g and 7g it will be plus 5g minus 7g so we will get to be as 11 minus 3 is 8 x so there is the razor and we got it 8x minus 6 y and minus use it this will be the vector b i've got these two vectors i find this this was the part a now in the second part i have to find the angle between a plus b and a for that first of all i have to find the magnitude the magnitude of a plus b will be equals to some of the square of all the value it can be equal to level square plus one square plus five square whole under root i will do it directly and it will be equal to wait a second under root 147 what i will get here is under root 147 i have to find now the magnitude of the second one it will be equals to again same thing it will be equal to 3 square plus 5 square plus 7 square whole under root and this will be equal to under root 83 the second thing i need is a dot b separated here now what is a dot b a dot b will be be equals to sorry it is not a dot b i have to be right here maybe i should go down here it will be a plus b i'll put in that dot a this is what i have to find here a plus b i have 11 x minus y plus 5 z this is the vector a plus b and the other vector is 3x plus 5 y plus 70 the dot product will be equals to 11 into 3 is 33 minus 5 plus 35 if you do the sum you will get it 63 now we are here at the final part which will get us the final answer what we have to find is the angle between them...

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