Problem 8 - Advanced Engineering Mathematics

Question

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b},(\mathbf{c}) \mathbf{a}-\mathbf{b},(\mathbf{d})\|\mathbf{a}+\mathbf{b}\|\), and \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=\langle 7,10\rangle, \mathbf{b}=\langle 1,2\rangle\)

Step-by-Step Solution

Answer

\(3\mathbf{a} = \langle 21, 30 \rangle\), \(\mathbf{a}+\mathbf{b} = \langle 8, 12 \rangle\), \(\mathbf{a}-\mathbf{b} = \langle 6, 8 \rangle\), \(\|\mathbf{a}+\mathbf{b}\| = 4\sqrt{13}\), \(\|\mathbf{a}-\mathbf{b}\| = 10\).

Step 1: Calculate 3a

First, multiply each component of vector \(\mathbf{a}\) by 3: \[3 \mathbf{a} = 3 \langle 7, 10 \rangle = \langle 3 \times 7, 3 \times 10 \rangle = \langle 21, 30 \rangle.\]

Step 2: Calculate a+b

Add the corresponding components of vectors \(\mathbf{a}\) and \(\mathbf{b}\):\[\mathbf{a} + \mathbf{b} = \langle 7, 10 \rangle + \langle 1, 2 \rangle = \langle 7+1, 10+2 \rangle = \langle 8, 12 \rangle.\]

Step 3: Calculate a-b

Subtract the corresponding components of vectors \(\mathbf{b}\) from \(\mathbf{a}\):\[\mathbf{a} - \mathbf{b} = \langle 7, 10 \rangle - \langle 1, 2 \rangle = \langle 7-1, 10-2 \rangle = \langle 6, 8 \rangle.\]

Step 4: Calculate |a+b|

Find the magnitude of the vector \(\mathbf{a} + \mathbf{b}\):\[\|\mathbf{a} + \mathbf{b}\| = \sqrt{8^2 + 12^2} = \sqrt{64 + 144} = \sqrt{208} = 4\sqrt{13}.\]

Step 5: Calculate |a-b|

Find the magnitude of the vector \(\mathbf{a} - \mathbf{b}\):\[\|\mathbf{a} - \mathbf{b}\| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10.\]

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector math. It involves multiplying a vector by a scalar (a real number). This operation scales the vector but does not change its direction. To perform scalar multiplication, you multiply each component of the vector by the scalar.For example, multiplying vector \( \mathbf{a} = \langle 7, 10 \rangle \) by 3 means calculating each component as follows:

• First component: \( 3 \times 7 = 21 \)
• Second component: \( 3 \times 10 = 30 \)

Thus, the result of the scalar multiplication is the vector \( 3\mathbf{a} = \langle 21, 30 \rangle \). This stretches the original vector by a factor of 3.

Vector Addition

Vector addition combines two or more vectors to produce a single vector. This operation is performed component-wise. To add two vectors together, simply add their corresponding components.Consider vectors \( \mathbf{a} = \langle 7, 10 \rangle \) and \( \mathbf{b} = \langle 1, 2 \rangle \):

• Add the first components: \( 7 + 1 = 8 \)
• Add the second components: \( 10 + 2 = 12 \)

Hence, the resultant vector \( \mathbf{a} + \mathbf{b} \) is \( \langle 8, 12 \rangle \). This operation effectively combines the effects of the two vectors into one.

Vector Subtraction

Vector subtraction involves taking one vector away from another, also done component-wise. This operation is similar to vector addition, but instead of adding, we subtract the corresponding components.For vectors \( \mathbf{a} = \langle 7, 10 \rangle \) and \( \mathbf{b} = \langle 1, 2 \rangle \), the subtraction will be:

• First component: \( 7 - 1 = 6 \)
• Second component: \( 10 - 2 = 8 \)

Therefore, the result of \( \mathbf{a} - \mathbf{b} \) is \( \langle 6, 8 \rangle \). This operation yields a vector pointing from the tip of \( \mathbf{b} \) to the tip of \( \mathbf{a} \).

Magnitude of a Vector

The magnitude of a vector is a measure of its length. It provides a scalar value that represents the total size of the vector. Calculating the magnitude involves using the Pythagorean theorem.For vector \( \mathbf{v} = \langle v_1, v_2 \rangle \), the magnitude \( \|\mathbf{v}\| \) is given by:\[\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}\]This calculation applies to vectors obtained from addition or subtraction as well. For \( \mathbf{a} + \mathbf{b} = \langle 8, 12 \rangle \):

• Calculate \( 8^2 + 12^2 = 64 + 144 = 208 \)
• The magnitude is \( \sqrt{208} = 4\sqrt{13} \)

And for \( \mathbf{a} - \mathbf{b} = \langle 6, 8 \rangle \):

• Calculate \( 6^2 + 8^2 = 36 + 64 = 100 \)
• The magnitude is \( \sqrt{100} = 10 \)

Magnitude helps to understand how "big" a vector is, regardless of its direction.