Problem 10 - Advanced Engineering Mathematics

Question

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\mathbf{i}+\mathbf{j}, \mathbf{b}=3 \mathbf{i}-2 \mathbf{j} \quad\)

Step-by-Step Solution

Answer

(a) \(-2\mathbf{i} + 8\mathbf{j}\); (b) \(-18\mathbf{i} + 7\mathbf{j}\).

Step 1: Write down the given vectors

We are given two vectors \( \mathbf{a} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{b} = 3\mathbf{i} - 2\mathbf{j} \). First, let's write these down so that we can perform vector operations on them.

Step 2: Calculate \(4\mathbf{a}\)

To find \(4\mathbf{a}\), we multiply each component of vector \(\mathbf{a}\) by 4. So, \( 4\mathbf{a} = 4(\mathbf{i} + \mathbf{j}) = 4\mathbf{i} + 4\mathbf{j} \).

Step 3: Calculate \(-2\mathbf{b}\)

To find \(-2\mathbf{b}\), we multiply each component of vector \(\mathbf{b}\) by -2. So, \(-2\mathbf{b} = -2(3\mathbf{i} - 2\mathbf{j}) = -6\mathbf{i} + 4\mathbf{j} \).

Step 4: Calculate \(4\mathbf{a} - 2\mathbf{b}\)

Now combine the results from Steps 2 and 3: \(4\mathbf{a} - 2\mathbf{b} = (4\mathbf{i} + 4\mathbf{j}) + (-6\mathbf{i} + 4\mathbf{j}) = (4 - 6)\mathbf{i} + (4 + 4)\mathbf{j} = -2\mathbf{i} + 8\mathbf{j} \).

Step 5: Calculate \(-3\mathbf{a}\)

To find \(-3\mathbf{a}\), we multiply each component of vector \(\mathbf{a}\) by -3. So, \(-3\mathbf{a} = -3(\mathbf{i} + \mathbf{j}) = -3\mathbf{i} - 3\mathbf{j} \).

Step 6: Calculate \(-5\mathbf{b}\)

To find \(-5\mathbf{b}\), we multiply each component of vector \(\mathbf{b}\) by -5. So, \(-5\mathbf{b} = -5(3\mathbf{i} - 2\mathbf{j}) = -15\mathbf{i} + 10\mathbf{j} \).

Step 7: Calculate \(-3\mathbf{a} - 5\mathbf{b}\)

Now combine the results from Steps 5 and 6: \(-3\mathbf{a} - 5\mathbf{b} = (-3\mathbf{i} - 3\mathbf{j}) + (-15\mathbf{i} + 10\mathbf{j}) = (-3 - 15)\mathbf{i} + (-3 + 10)\mathbf{j} = -18\mathbf{i} + 7\mathbf{j} \).

Key Concepts

Vector Addition

Vector addition is a foundational operation in vector mathematics. It involves combining two vectors to create a new vector that represents the cumulative effect of both. Imagine each vector as an arrow pointing in space. To add them, you position the start of the second arrow at the tip of the first arrow. The new vector, or resultant vector, is drawn from the start of the first arrow to the tip of the second arrow. This process is quite visual.

In algebraic terms, vector addition involves adding the corresponding components of each vector. For example, when adding vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), the sum is:

• The i-component: \(a_1 + b_1\)
• The j-component: \(a_2 + b_2\)

Thus, the resultant vector is \((a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j}\). This simple rule makes vector addition straightforward and reliable for combining forces, velocities, and other directional quantities.

Scalar Multiplication

Scalar multiplication is another key operation in vector math, which involves multiplying a vector by a scalar, or a single number. This operation scales the vector by the specified amount, altering its magnitude but not its direction.

To multiply a vector by a scalar, you enlarge or shrink each component of the vector by the scalar. For instance, given vector \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} \) and a scalar \( k \), the resulting vector after scalar multiplication is:

• For the i-component: \( kv_1 \)
• For the j-component: \( kv_2 \)

Thus, the new vector is \( kv_1\mathbf{i} + kv_2\mathbf{j} \).

Scalar multiplication can be used to change the scale of vectors in physics and engineering applications, such as adjusting velocity and force by a certain factor. This technique simplifies calculations while preserving the original direction of the vector.

Vector Subtraction

Vector subtraction, like vector addition, is essential in comparing and calculating the difference between two vector quantities. It involves finding another vector that shows how to go from the tip of one vector, typically called \( \mathbf{b} \), to the tip of another vector, \( \mathbf{a} \).

The process is similar to adding vectors, but instead, we reverse the direction of the vector being subtracted and then add it. Mathematically, subtracting a vector \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \) from vector \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) involves:

• For the i-component: \( a_1 - b_1 \)
• For the j-component: \( a_2 - b_2 \)

This results in the vector \((a_1 - b_1)\mathbf{i} + (a_2 - b_2)\mathbf{j}\).

Vector subtraction is crucial in mechanics, physics, and engineering, often used to find relative velocities or displacements between objects. It offers a straightforward way to deduce one vector's influence when offset against another.