Calculating Vector Modulus And Operations

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In vector algebra, the modulus or magnitude of a vector represents its length. It is a scalar quantity and is calculated using the components of the vector. Understanding how to find the modulus of a vector is fundamental in various applications, including physics, engineering, and computer graphics. This article will guide you through calculating the modulus of given vectors and performing operations on vectors.

a. Finding the Modulus of $2i - 5j + 7k$

To determine the modulus of the vector $2i - 5j + 7k$, we apply the formula for the magnitude of a vector in three-dimensional space. The vector's components are 2, -5, and 7 along the i, j, and k axes, respectively. The modulus, often denoted as |v|, is calculated as the square root of the sum of the squares of these components. This calculation provides the length of the vector, a crucial piece of information for many applications.

Step-by-Step Calculation

  1. Identify the components: The components of the vector are 2, -5, and 7.
  2. Square each component:
    • 22=42^2 = 4
    • (−5)2=25(-5)^2 = 25
    • 72=497^2 = 49
  3. Sum the squares: 4+25+49=784 + 25 + 49 = 78
  4. Take the square root of the sum: $\sqrt{78}$

Thus, the modulus of the vector $2i - 5j + 7k$ is $\sqrt{78}$. This value represents the length of the vector in three-dimensional space. The modulus is always a non-negative real number, reflecting the distance from the origin to the point represented by the vector. Knowing the modulus helps in normalizing vectors, finding unit vectors, and understanding the physical significance of vector quantities in various fields like physics and engineering. For instance, in physics, it can represent the magnitude of a force or velocity vector.

b. Finding the Modulus of $3i + 8j - k$

Next, let's calculate the modulus of the vector $3i + 8j - k$. Similar to the previous example, we will use the formula for the magnitude of a vector in three-dimensional space. The components of this vector are 3, 8, and -1 along the i, j, and k axes, respectively. The modulus provides the length or magnitude of the vector, which is essential for various applications, including vector normalization and calculations involving vector quantities in physics and engineering.

Step-by-Step Calculation

  1. Identify the components: The components of the vector are 3, 8, and -1.
  2. Square each component:
    • 32=93^2 = 9
    • 82=648^2 = 64
    • (−1)2=1(-1)^2 = 1
  3. Sum the squares: 9+64+1=749 + 64 + 1 = 74
  4. Take the square root of the sum: $\sqrt{74}$

Therefore, the modulus of the vector $3i + 8j - k$ is $\sqrt{74}$. This result indicates the length of the vector in three-dimensional space. The modulus is a scalar quantity and is always non-negative, representing the distance from the origin to the vector's endpoint. Understanding how to calculate the modulus is crucial for working with vectors, especially in physics, where it can represent the magnitude of physical quantities like force or velocity. This calculation is a fundamental step in many vector-related problems and applications.

Given Vectors

Consider three vectors a, b, and c defined as follows:

  • a=2i+3j−4ka = 2i + 3j - 4k

  • b=5i+4j+7kb = 5i + 4j + 7k

  • c=−6i+2j−kc = -6i + 2j - k

We will now perform vector operations to find the results of $2a + 3b - c$ and $a - 2b + 4c$. These operations involve scalar multiplication and vector addition/subtraction, which are essential skills in vector algebra. Understanding these operations is critical for solving problems in physics, engineering, and computer graphics, where vectors are used to represent forces, velocities, and spatial relationships.

i. Finding $2a + 3b - c$

To calculate $2a + 3b - c$, we first need to perform scalar multiplication on vectors a and b, and then carry out vector addition and subtraction. Scalar multiplication involves multiplying each component of the vector by the scalar, while vector addition and subtraction involve adding or subtracting corresponding components of the vectors. This process ensures that we correctly combine the vectors according to the given operation.

Step-by-Step Calculation

  1. Scalar multiplication:

    • 2a=2(2i+3j−4k)=4i+6j−8k2a = 2(2i + 3j - 4k) = 4i + 6j - 8k

    • 3b=3(5i+4j+7k)=15i+12j+21k3b = 3(5i + 4j + 7k) = 15i + 12j + 21k

  2. Vector addition and subtraction:

    • 2a+3b−c=(4i+6j−8k)+(15i+12j+21k)−(−6i+2j−k)2a + 3b - c = (4i + 6j - 8k) + (15i + 12j + 21k) - (-6i + 2j - k)

  3. Combine like terms:

    • =(4+15+6)i+(6+12−2)j+(−8+21+1)k= (4 + 15 + 6)i + (6 + 12 - 2)j + (-8 + 21 + 1)k

    • =25i+16j+14k= 25i + 16j + 14k

Therefore, $2a + 3b - c = 25i + 16j + 14k$. This resulting vector is the linear combination of the given vectors a, b, and c, and it represents a new vector in three-dimensional space. The components of this vector indicate its direction and magnitude along the i, j, and k axes. This type of calculation is fundamental in many areas of physics and engineering, where vector operations are used to analyze forces, velocities, and other physical quantities.

ii. Finding $a - 2b + 4c$

Now, let's compute $a - 2b + 4c$. This operation involves similar steps to the previous calculation, including scalar multiplication and vector addition/subtraction. We'll first multiply vectors b and c by the scalars -2 and 4, respectively, and then perform the vector addition and subtraction. This process ensures that we accurately combine the vectors according to the given expression.

Step-by-Step Calculation

  1. Scalar multiplication:

    • −2b=−2(5i+4j+7k)=−10i−8j−14k-2b = -2(5i + 4j + 7k) = -10i - 8j - 14k

    • 4c=4(−6i+2j−k)=−24i+8j−4k4c = 4(-6i + 2j - k) = -24i + 8j - 4k

  2. Vector addition and subtraction:

    • a−2b+4c=(2i+3j−4k)+(−10i−8j−14k)+(−24i+8j−4k)a - 2b + 4c = (2i + 3j - 4k) + (-10i - 8j - 14k) + (-24i + 8j - 4k)

  3. Combine like terms:

    • =(2−10−24)i+(3−8+8)j+(−4−14−4)k= (2 - 10 - 24)i + (3 - 8 + 8)j + (-4 - 14 - 4)k

    • =−32i+3j−22k= -32i + 3j - 22k

Thus, $a - 2b + 4c = -32i + 3j - 22k$. This vector is another linear combination of the vectors a, b, and c. The components -32, 3, and -22 represent the vector's direction and magnitude along the i, j, and k axes, respectively. Such calculations are essential in various scientific and engineering applications, where vector operations are used to model and analyze complex systems and phenomena.

In summary, we have demonstrated how to find the modulus of vectors and perform vector operations such as scalar multiplication, addition, and subtraction. The modulus of a vector gives its length, which is a fundamental property in many applications. Vector operations allow us to combine vectors in various ways, which is crucial in fields like physics, engineering, and computer graphics. The ability to perform these calculations accurately is essential for anyone working with vector quantities. Understanding these concepts provides a solid foundation for more advanced topics in linear algebra and vector calculus.