Problem 21 - Advanced Engineering Mathematics

Question

Determine which of the following vectors are parallel to \(\mathbf{a}=4 \mathbf{i}+6 \mathbf{j}\) (a) \(-4 \mathbf{i}-6 \mathbf{j}\) (b) \(-\mathbf{i}-\frac{3}{2} \mathbf{j}\) (c) \(10 \mathbf{i}+15 \mathbf{j}\) (d) \(2(\mathbf{i}-\mathbf{j})-3\left(\frac{1}{2} \mathbf{i}-\frac{5}{12} \mathbf{j}\right)\) (e) \(8 \mathbf{i}+12 \mathbf{j}\) (f) \((5 \mathbf{i}+\mathbf{j})-(7 \mathbf{i}+4 \mathbf{j})\)

Step-by-Step Solution

Answer

Vectors (a), (c), (e), and (f) are parallel to \(\mathbf{a}\).

Step 1: Understanding Vector Parallelism

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are parallel if there exists a scalar \(k\) such that \(\mathbf{v} = k\mathbf{u}\). This means each component of \(\mathbf{v}\) is a scalar multiple of the corresponding component of \(\mathbf{u}\). In this problem, we need to find vectors that are scalar multiples of \(\mathbf{a} = 4\mathbf{i} + 6\mathbf{j}\).

Step 2: Check Vector (a)

Vector (a) is \(-4\mathbf{i} - 6\mathbf{j}\). For these vectors to be parallel, \(-4\) should be equal to \(k \times 4\) and \(-6\) should be equal to \(k \times 6\). Solving \(k = -1\), we find that vector (a) is parallel to \(\mathbf{a}\).

Step 3: Check Vector (b)

Vector (b) is \(-\mathbf{i} - \frac{3}{2}\mathbf{j}\). For parallelism, \(-1\) should be equal to \(k \times 4\) and \(-\frac{3}{2}\) to \(k \times 6\). Solving these equations, no consistent \(k\) satisfies both equations, so vector (b) is not parallel to \(\mathbf{a}\).

Step 4: Check Vector (c)

Vector (c) is \(10\mathbf{i} + 15\mathbf{j}\). We solve \(k \times 4 = 10\) which gives \(k = \frac{5}{2}\), and \(k \times 6 = 15\) checks out, so vector (c) is parallel to \(\mathbf{a}\).

Step 5: Check Vector (d)

Simplify vector (d). It is \(2\mathbf{i} - 2\mathbf{j} - 3 \Big( \frac{1}{2}\mathbf{i} - \frac{5}{12}\mathbf{j} \Big) = \mathbf{i} - \frac{7}{4}\mathbf{j}\). For \(\mathbf{i} - \frac{7}{4}\mathbf{j}\) to be parallel, neither coordinate works with a consistent \(k\), thus vector (d) is not parallel to \(\mathbf{a}\).

Step 6: Check Vector (e)

Vector (e) is \(8\mathbf{i} + 12\mathbf{j}\). Solving for \(k\), we have \(k = 2\) for both components, so vector (e) is parallel to \(\mathbf{a}\).

Step 7: Check Vector (f)

Vector (f) simplifies to \( (5\mathbf{i} + \mathbf{j}) - (7\mathbf{i} + 4\mathbf{j}) = -2\mathbf{i} - 3\mathbf{j} \). For parallelism, solve \(-2 = k \times 4\) and \(-3 = k \times 6\), yielding \(k = -\frac{1}{2}\). They are consistent, so vector (f) is parallel to \(\mathbf{a}\).

Key Concepts

Scalar Multiplication

Scalar multiplication in vectors involves multiplying each component of a vector by a scalar value. This concept is key to understanding parallel vectors. To determine if two vectors are parallel, we explore if one vector is a scalar multiple of the other. For example, if you have a vector \(\mathbf{a} = 4\mathbf{i} + 6\mathbf{j}\), any vector parallel to \(\mathbf{a}\) can be expressed as \(k(4\mathbf{i} + 6\mathbf{j})\), where \(k\) is a scalar.

• This means both components of the new vector must have the same ratio \(k\) with \(4\) and \(6\).
• For example, to check if a vector \(-4\mathbf{i} - 6\mathbf{j}\) is parallel, solve for \(k\) such that \(-4 = k \times 4\) and \(-6 = k \times 6\). Solving these gives \(k = -1\), confirming parallelism.

If a consistent scalar \(k\) does not exist for both components, then the vectors are not parallel.

Coordinate Geometry

Coordinate geometry helps us visualize vectors on a plane using \(\mathbf{i}\) and \(\mathbf{j}\) components.

• The \(\mathbf{i}\) and \(\mathbf{j}\) notations represent the x-direction and y-direction of the vector in a 2D coordinate plane.
• Lay down the x-y plane and place the tail of the vector at the origin to visualize its direction and magnitude.
• For a vector \(\mathbf{a} = 4\mathbf{i} + 6\mathbf{j}\), the point on the plane it points to is (4, 6).

Using coordinate geometry, you can easily see the position of vectors, making it straightforward to check parallelism since parallel vectors will align along the same line or direction in the coordinate plane.

Vector Components

Vectors are often broken down into components that represent their influence in different directions on the plane. Each vector is the sum of its component vectors along the coordinate axes.

• The \(\mathbf{i}\)-component indicates how far the vector stretches along the x-axis, and the \(\mathbf{j}\)-component shows the range along the y-axis.
• For \(\mathbf{a} = 4\mathbf{i} + 6\mathbf{j}\), the components \(4\mathbf{i}\) and \(6\mathbf{j}\) can be visualized as the horizontal and vertical sides of a right triangle, with the vector as the hypotenuse.

Understanding these components facilitates determining vector equality, direction, and parallelism. When a scalar multiplication satisfies both components evenly, the vectors are indeed parallel. This detailed breakdown of vectors into components is crucial for verifying simple numerical conditions to assess parallelism.