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Chapter 10: Problem 4

Compute, \(a+b, a-2 b, 3 a\) and \(\|5 b-2 a\|.\) $$a=(3,-2), b=\langle 2,0\rangle$$

Short Answer

Expert verified
The computed vectors are \(a + b = (5,-2)\), \(a - 2b = (-1,-2)\), \(3a = (9,-6)\), and the magnitude of \(5b - 2a\) is \( ||5b - 2a||= \sqrt{32}\).

Step by step solution

01

Compute the addition of vectors a and b

To add two vectors, just add their corresponding components (i and j components). So, given \(a=(3,-2)\) and \(b=\langle 2,0\rangle\), we have \(a + b = (3+2, -2+0) = (5,-2)\).
02

Compute the subtraction of two times vector b from vector a

The subtraction of two vectors involves multiplying the vectors by their coefficients and then adding the results. So, \(a - 2b = a + (-2)*b = (3,-2) + (-2)*(2,0) = (3,-2) + (-4,0) = (3-4,-2+0) = (-1,-2)\).
03

Compute three times vector a

Multiplication of vector a by the scalar 3 is performed by multiplying each component of a by 3. So, \(3a = 3*(3,-2) = (9,-6)\).
04

Compute the magnitude of the vector \(5b - 2a\)

First, find the vector \(5b - 2a = 5*(2,0) - 2*(3,-2) = (10,0) - (6,-4) = (4,4) \). Then, calculate the magnitude of this vector using the formula \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), where (x_1,y_1) and (x_2,y_2) are the components of 5b-2a. So, \( ||5b - 2a|| = \sqrt{(4 - 0)^2 + (4 - 0)^2} = \sqrt{16 + 16} = \sqrt{32}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Addition
Vector addition is the process of taking two or more vectors and combining them into a single vector, known as the resultant vector. This involves adding the corresponding components of each vector together. For example, if we have vectors \(a=(3,-2)\) and \(b=\langle 2,0\rangle\), the addition is performed by:
  • Adding the first components: \(3 + 2 = 5\)
  • Adding the second components: \(-2 + 0 = -2\)
Thus, the result of adding vectors \(a\) and \(b\) is \((5, -2)\). This new vector, \(\mathbf{R}\), represents the combined effect of both vectors' contributions.
Vector Subtraction
Vector subtraction involves finding the difference between two vectors. To do this, you subtract one vector from another, which is similar to adding the negative of the vector being subtracted. If we have a vector \(a = (3, -2)\) and we want to subtract \(2\) times vector \(b = \langle 2,0 \rangle\), we first scale \(b\) by multiplying each component by \(2\):
  • \(2 \times 2 = 4\)
  • \(2 \times 0 = 0\)
This gives us \((4, 0)\). Then we calculate:
  • \(3 - 4 = -1\)
  • \(-2 - 0 = -2\)
Therefore, \(a - 2b\) results in \((-1, -2)\). This operation effectively shifts the vector \(a\) by the vector \(-2b\), resulting in a new vector.
Scalar Multiplication
Scalar multiplication of a vector involves multiplying each component of a vector by a scalar (a real number). For example, if we multiply vector \(a = (3, -2)\) by the scalar \(3\), we apply the multiplication to each component individually:
  • \(3 \times 3 = 9\)
  • \(3 \times -2 = -6\)
So, multiplying vector \(a\) by \(3\) gives us the new vector \((9, -6)\). This process effectively scales the vector, maintaining its direction but increasing its magnitude. Scalar multiplication is useful for scaling vectors up or down to a desired size and is essential in many vector operations.
Vector Magnitude
The magnitude of a vector represents its length or size, without considering its direction. It is calculated using the Pythagorean theorem. For a vector \(v = (x, y)\), the magnitude \(||v||\) is:\[||v|| = \sqrt{x^2 + y^2}\]In our case, to find the magnitude of \(5b - 2a\), we first compute the vector \(5b - 2a\), which is \((4, 4)\). Then, apply the formula:
  • Calculate \(4^2 = 16\)
  • Add \(16 + 16 = 32\)
  • Take the square root: \(\sqrt{32} = 4\sqrt{2}\)
The magnitude \(4\sqrt{2}\) represents the length of the vector \(5b - 2a\). Understanding vector magnitude is crucial for determining the vector's influence or strength in a given context.

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Most popular questions from this chapter

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