Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector.$$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \mathbf{v}=-\mathbf{i}+\mathbf{j}$$

Step-by-Step Solution

Answer

(a) The resultant vector is \(\mathbf{i}\), (b) The resultant vector is \(3\mathbf{i} - 2\mathbf{j}\), (c) The resultant vector is \(7\mathbf{i} - 5\mathbf{j}\), (d) The resultant vector is \(8\mathbf{i} - 3.5\mathbf{j}\)

Step 1: Calculate \(\mathbf{u}+\mathbf{v}\)

Add corresponding components of vectors \(\mathbf{u}\) and \(\mathbf{v}\). Hence, \(\mathbf{u}+\mathbf{v} = (2 - 1) \mathbf{i}+ (-1 + 1) \mathbf{j} = \mathbf{i}\). The resultant vector is \(\mathbf{i}\).

Step 2: Calculate \(\mathbf{u}-\mathbf{v}\)

Subtract corresponding components of vectors \(\mathbf{v}\) from \(\mathbf{u}\). Hence, \(\mathbf{u}-\mathbf{v} = (2 - (-1)) \mathbf{i}+ (-1 - 1) \mathbf{j} = 3\mathbf{i} - 2\mathbf{j}\). The resultant vector is \(3\mathbf{i} - 2\mathbf{j}\).

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

For this step, first multiply \(\mathbf{u}\) by 2, and \(\mathbf{v}\) by 3, and then subtract the corresponding components. Hence, \(2\mathbf{u}-3\mathbf{v} = (2*2 - 3*(-1)) \mathbf{i}+ (2*-1 - 3*1) \mathbf{j} = 7\mathbf{i} - 5\mathbf{j}\). The resultant vector is \(7\mathbf{i} - 5\mathbf{j}\).

Step 4: Calculate \(\frac{1}{2}\mathbf{v}+4\mathbf{u}\)

For this step, first multiply \(\mathbf{v}\) by \(\frac{1}{2}\), and \(\mathbf{u}\) by 4, and then add the corresponding components. Hence, \(\frac{1}{2}\mathbf{v}+4\mathbf{u} = (4*2 + \frac{1}{2}*(-1)) \mathbf{i}+ (4*-1 + \frac{1}{2}*1) \mathbf{j} =8\mathbf{i} - 3.5\mathbf{j}\). The resultant vector is \(8\mathbf{i} - 3.5\mathbf{j}\).

Step 5: Sketching the vectors

Sketching is done by plotting each vector on a graph. The x coordinate of a vector corresponds to the coefficient of \(i\), and the y coordinate corresponds to the coefficient of \(j\) and then drawing a arrow from origin to the point (x, y).

Key Concepts

Understanding Vector Addition

Vector addition is a fundamental operation where two vectors are combined to form a single resultant vector. To perform vector addition, you simply add the corresponding components of the vectors involved. For example, given vectors \(\mathbf{u} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{v} = -\mathbf{i} + \mathbf{j}\), their sum \(\mathbf{u} + \mathbf{v}\) is calculated as follows:

• For the x-component: Add \(2\) from \(\mathbf{u}\) and \(-1\) from \(\mathbf{v}\), resulting in \(2 - 1 = 1\).
• For the y-component: Add \(-1\) from \(\mathbf{u}\) and \(1\) from \(\mathbf{v}\), resulting in \(-1 + 1 = 0\).

Thus, the resultant vector from adding \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathbf{i}\). This shows how vector addition can be understood as a simple combination of directions and magnitudes.
Always make sure to handle each component separately and remember that direction matters in vector addition.

Exploring Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which is a single real number. This operation stretches or shrinks the vector by the multiplier's magnitude, affecting both direction and length. Given \(\mathbf{u} = 2\mathbf{i} - \mathbf{j}\) and a scalar \(2\), the multiplication \(2\mathbf{u}\) is carried out by multiplying each component:

• For the x-component: Multiply \(2\) by \(2\), giving \(4\mathbf{i}\).
• For the y-component: Multiply \(-1\) by \(2\), resulting in \(-2\mathbf{j}\).

The resultant vector becomes \(4\mathbf{i} - 2\mathbf{j}\). Scalar multiplication is key when preparing a vector for further operations, such as adding to or subtracting from another vector, to achieve a specific resultant vector.

Breaking Down Vector Subtraction

Vector subtraction differs from addition by involving the subtraction of the corresponding components of vectors. If we subtract vector \(\mathbf{v}\) from \(\mathbf{u}\), expressed as \(\mathbf{u} - \mathbf{v}\), it proceeds by changing the sign of each component of \(\mathbf{v}\) and adding:

• For the x-component: Compute \(2 - (-1) = 3\mathbf{i}\), which involves adding \(1\mathbf{i}\).
• For the y-component: Compute \(-1 - 1 = -2\mathbf{j}\).

This gives the resultant vector \(3\mathbf{i} - 2\mathbf{j}\). Vector subtraction is useful for finding the difference or directional change between two vectors. It essentially shows how one vector relates to another in terms of direction and magnitude.

Illustrating Resultant Vector Sketching

Sketching a resultant vector is a visual representation of mathematical vector operations on a plane. Each vector is depicted starting from the origin with an arrow pointing to the coordinates determined by its components. For instance, if a vector has components \(x\mathbf{i} + y\mathbf{j}\), this translates to plotting the point (x, y) and drawing an arrow from the start point (usually the origin) to this location.
To sketch the resultant vector \(8\mathbf{i} - 3.5\mathbf{j}\):

• Point (8, -3.5): Start at the origin (0,0), move 8 units along the x-axis.
• Then move down 3.5 units along the y-axis.

Connecting these dots with an arrow gives the direction and length of the resultant vector. Vector sketching is essential for a visual understanding of vector operations, aiding in comprehending the geometry involved in vector mathematics.