Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluating an Expression Evaluate the expression. $$\frac{1}{2}\left(\left[\begin{array}{ccc} 3 & -2 & 4 & 0 \end{array}\right]-\left[\begin{array}{cccc} 10 & -6 & -18 & 9 \end{array}\right]\right)$$

Step-by-Step Solution

Answer

The final answer is [-3.5, 2, 11, -4.5].

Step 1: Subtraction of Vectors

To subtract two vectors, subtract corresponding components. So, [(3, -2, 4, 0)] - [(10, -6, -18, 9)] becomes [3-10, -2-(-6), 4-(-18), 0-9] = [-7, 4, 22, -9].

Step 2: Multiplying by a Scalar

To multiply a vector by a scalar, each component of the vector should be multiplied by the scalar. Therefore, 1/2 * [-7, 4, 22, -9] = [-7/2, 4/2, 22/2, -9/2].

Step 3: Simplifying the vector

To simplify, perform each operation: [-7/2, 4/2, 22/2, -9/2] = [-3.5, 2, 11, -4.5].

Key Concepts

Vector Subtraction

Vector subtraction is the process of subtracting each corresponding component of one vector from another. In our example, we have two vectors:

• \([3, -2, 4, 0]\)
• \([10, -6, -18, 9]\)

To subtract these vectors, you need to look at each matching element from both vectors. Start with the first component in each vector: subtract 10 from 3, giving you -7 for the first component of the resulting vector.
Repeat this process for the remaining components:

• For the second component: \(-2 - (-6) = 4\)
• For the third component: \(4 - (-18) = 22\)
• For the fourth component: \(0 - 9 = -9\)

The new vector after subtraction is \([-7, 4, 22, -9]\). Just remember, subtract each pair of corresponding components to get your result. It's almost like subtraction in arithmetic, just spread out across multiple dimensions!

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a constant number). In the expression given, our scalar is \(\frac{1}{2}\).
This means we take each component of the vector \([-7, 4, 22, -9]\) and multiply it by \(\frac{1}{2}\). Let's go through each component step-by-step:

• First component: \(-7 \times \frac{1}{2} = -3.5\)
• Second component: \(4 \times \frac{1}{2} = 2\)
• Third component: \(22 \times \frac{1}{2} = 11\)
• Fourth component: \(-9 \times \frac{1}{2} = -4.5\)

The resulting vector is \([-3.5, 2, 11, -4.5]\).
Scalar multiplication simply scales the vector, uniformly transforming its magnitude based on the scalar, without affecting its direction, unless the scalar is negative, in which case the vector's direction also reverses.

Simplifying Vectors

Simplifying vectors often refers to performing and finalizing arithmetic operations and ensuring the results are expressed in their simplest form. After performing vector subtraction and scalar multiplication, you end with a new vector ready to be interpreted, but sometimes these components can further be simplified, as in changing fractions to decimals if needed.
In our calculation, we've arrived at \([-3.5, 2, 11, -4.5]\), which is already simplified. Here, simplification entailed ensuring the operations were correctly performed and reducing fractions where possible, like converting a fraction such as \(-\frac{7}{2}\) directly into \(-3.5\).
Always double-check your vector components after simplifying to confirm they represent your final, precise answer. Simplifying correctly is crucial as it maintains the integrity of your mathematical results!