Problem 8

Question

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

Step-by-Step Solution

Verified

Answer

(a) (21, 30), (b) (8, 12), (c) (6, 8), (d) $4\sqrt{13}$, (e) 10

1Step 1: Calculate 3a

To find $3\mathbf{a}$, you multiply each component of $\mathbf{a}=(7,10)$ by 3. \[3\mathbf{a} = 3 \times (7, 10) = (21, 30)\]

2Step 2: Calculate a+b

To find $\mathbf{a} + \mathbf{b}$, add the corresponding components of $\mathbf{a}$ and $\mathbf{b}$. \[\mathbf{a} + \mathbf{b} = (7 + 1, 10 + 2) = (8, 12)\]

3Step 3: Calculate a-b

To find $\mathbf{a} - \mathbf{b}$, subtract the corresponding components of $\mathbf{b}$ from $\mathbf{a}$. \[\mathbf{a} - \mathbf{b} = (7 - 1, 10 - 2) = (6, 8)\]

4Step 4: Calculate the magnitude of a+b

The magnitude of $\mathbf{a} + \mathbf{b}$ is calculated using the formula $\|\mathbf{c}\| = \sqrt{x^2 + y^2}$ for a vector $\mathbf{c}=(x, y)$. Here, $\mathbf{c} = (8, 12)$. \[\|\mathbf{a} + \mathbf{b}\| = \sqrt{8^2 + 12^2} = \sqrt{64 + 144} = \sqrt{208} = 2\sqrt{52} = 2\sqrt{4 \times 13} = 4\sqrt{13}\]

5Step 5: Calculate the magnitude of a-b

The magnitude of $\mathbf{a} - \mathbf{b}$ is calculated in the same way. \[\|\mathbf{a} - \mathbf{b}\| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\]

Key Concepts

Scalar MultiplicationVector AdditionVector SubtractionMagnitude of a Vector

Scalar Multiplication

Scalar multiplication in vector algebra involves multiplying a vector by a scalar (a constant number). This operation magnifies or shrinks the vector while maintaining its direction. To perform scalar multiplication, you multiply each component of the vector individually by the scalar value.

For example, given the vector $ \mathbf{a} = (7, 10) $ and the scalar $ 3 $, you would calculate $ 3\mathbf{a} $ as follows:

Multiply the first component: $ 3 \times 7 = 21 $
Multiply the second component: $ 3 \times 10 = 30 $

Thus, the resulting vector after scalar multiplication is $ 3\mathbf{a} = (21, 30) $.

This shows how the vector's size is scaled, but its direction remains unchanged.

Vector Addition

Vector addition is the process of adding two vectors to form a new vector, which translates the concept of 'combining' the effects of two separate vectors.

To perform vector addition, you add the corresponding components of each vector. Given vectors $ \mathbf{a} = (7, 10) $ and $ \mathbf{b} = (1, 2) $, find $ \mathbf{a} + \mathbf{b} $ as follows:

Add the first components: $ 7 + 1 = 8 $
Add the second components: $ 10 + 2 = 12 $

This results in the new vector $ \mathbf{a} + \mathbf{b} = (8, 12) $.

Vector addition is fundamental in physics and engineering, particularly when calculating the resultant vector of concurrent forces.

Vector Subtraction

Vector subtraction involves removing the influence of one vector from another, effectively shifting the resulting vector in the direction opposite to the one being subtracted.

To subtract vectors, subtract the components of the second vector from the first. For vectors $ \mathbf{a} = (7, 10) $ and $ \mathbf{b} = (1, 2) $, subtract as follows:

Subtract the first components: $ 7 - 1 = 6 $
Subtract the second components: $ 10 - 2 = 8 $

The resulting vector after subtraction is $ \mathbf{a} - \mathbf{b} = (6, 8) $.

This operation is useful in various applications, such as determining the relative position or velocity of an object.

Magnitude of a Vector

The magnitude of a vector represents its length and is a crucial concept in understanding how much 'force' or 'distance' the vector conveys.

To find the magnitude of a vector $ \mathbf{v} = (x, y) $, use the formula: \[ \|\mathbf{v}\| = \sqrt{x^2 + y^2} \]

For the vector $ \mathbf{c} = (8, 12) $, corresponding to $ \mathbf{a} + \mathbf{b} $, calculate its magnitude:

Compute $ 8^2 + 12^2 = 64 + 144 = 208 $
Take the square root: $ \sqrt{208} = 2\sqrt{52} = 4\sqrt{13} $

Similarly, for $ \mathbf{a} - \mathbf{b} = (6, 8) $, evaluate: