Scalar Vector Multiplication

Definition of Scalar Multiplication of Vectors

Scalar multiplication of a vector is an operation involving multiplication between a real number (scalar) and a vector $\vec{v}$. The result of this multiplication is a new vector with a length modified according to the scalar value, while its direction may remain the same or opposite depending on the sign of the scalar.

If $k$ is a real number (scalar) and $\vec{v}$ is a vector, then the scalar multiplication of a vector is denoted as $k \cdot \vec{v}$ and results in a new vector.

Properties of Scalar Multiplication of Vectors

Scalar multiplication of vectors has several important properties:

1. If $k > 0$ (positive), then the resulting vector has the same direction as the original vector.
2. If $k < 0$ (negative), then the resulting vector has a direction opposite to the original vector.
3. If $k = 0$, then the resulting vector is a zero vector.
4. The magnitude (length) of the resulting vector is $|k|$ times the magnitude of the original vector.

Representation of Scalar Multiplication of Vectors

Geometrically

Geometrically, scalar multiplication of a vector changes the length (magnitude) of the vector by $|k|$ times. The direction of the vector depends on the sign of $k$:

• If $k > 0$, the direction of the vector remains unchanged
• If $k < 0$, the direction of the vector is opposite to the original vector $|k \cdot \vec{v}| = |k| \cdot |\vec{v}|$

Algebraically

If $\vec{v} = (v_1, v_2, v_3)$ is a vector in 3-dimensional space, then:

In unit vector notation:

Examples of Scalar Multiplication of Vectors

Example 1

Given the vector $\vec{a} = 2\vec{i} + 3\vec{j} + 5\vec{k}$. Determine the result of multiplication $2\vec{a}$.

Solution:

Example 2

Given the vector $\vec{v} = (4, -2, 3)$. Determine the result of $-3\vec{v}$.

Solution:

Note that the direction of the resulting vector is opposite to the original vector because the scalar is negative.

Applications of Scalar Multiplication of Vectors

Scalar multiplication of vectors has many applications in physics and mathematics, such as:

1. Force and Acceleration: If an object with mass $m$ experiences acceleration $\vec{a}$, then the force acting on the object is $\vec{F} = m\vec{a}$.

2. Velocity: If an object moves with velocity $\vec{v}$ for a time $t$, then the displacement of the object is $\vec{s} = t\vec{v}$.

3. Scaling in Computer Graphics: To change the size of objects in computer graphics, the coordinates of points on the object are multiplied by a scale factor.

Practice Problems

1. Given the vector $\vec{a} = 3\vec{i} - 4\vec{j} + 2\vec{k}$. Determine the result of $-2\vec{a}$.

2. Vectors $\vec{u} = (2, -5, 1)$ and $\vec{v} = (-4, 3, 6)$. Determine the vector $2\vec{u} - 3\vec{v}$.

3. Given the vector $\overrightarrow{BR} = 3.4 \text{ cm}$. If $\overrightarrow{BU} = 0.65 \cdot \overrightarrow{BR}$ and $\overrightarrow{UR} = 0.35 \cdot \overrightarrow{BR}$, prove that all three vectors have the same direction.

4. Vector $\vec{p}$ has a length of 5 units and vector $\vec{q} = 3\vec{p}$. Determine the length of vector $\vec{q}$.

5. Given points $A(2, 3, -1)$, $B(5, -2, 4)$, and $C$ lies on the line passing through $A$ and $B$ such that $\overrightarrow{AC} = 2\overrightarrow{AB}$. Determine the coordinates of point $C$.

Answer Key

Problem 1

Given the vector $\vec{a} = 3\vec{i} - 4\vec{j} + 2\vec{k}$. Determine the result of $-2\vec{a}$.

Solution:

Therefore, the result of $-2\vec{a}$ is $-6\vec{i} + 8\vec{j} - 4\vec{k}$.

Problem 2

Vectors $\vec{u} = (2, -5, 1)$ and $\vec{v} = (-4, 3, 6)$. Determine the vector $2\vec{u} - 3\vec{v}$.

Solution:

Therefore, the vector $2\vec{u} - 3\vec{v}$ is $(16, -19, -16)$ or $16\vec{i} - 19\vec{j} - 16\vec{k}$.

Problem 3

Given the vector $\overrightarrow{BR} = 3.4 \text{ cm}$. If $\overrightarrow{BU} = 0.65 \cdot \overrightarrow{BR}$ and $\overrightarrow{UR} = 0.35 \cdot \overrightarrow{BR}$, prove that all three vectors have the same direction.

Solution: To prove that all three vectors have the same direction, we need to show that they are positive scalar multiples of the same vector.

We know:

• $\overrightarrow{BU} = 0.65 \cdot \overrightarrow{BR}$
• $\overrightarrow{UR} = 0.35 \cdot \overrightarrow{BR}$

Let's check if $\overrightarrow{BU} + \overrightarrow{UR} = \overrightarrow{BR}$:

This result shows that $\overrightarrow{BU} + \overrightarrow{UR} = \overrightarrow{BR}$, which aligns with the vector addition law for collinear points B, U, and R.

Since $\overrightarrow{BU} = 0.65 \cdot \overrightarrow{BR}$ and $\overrightarrow{UR} = 0.35 \cdot \overrightarrow{BR}$, where the scalar factors are positive ($0.65$ and $0.35$), all three vectors have the same direction. Positive scalar factors mean that these vectors point in the same direction as the reference vector $\overrightarrow{BR}$.

Therefore, it is proven that the three vectors $\overrightarrow{BR}$, $\overrightarrow{BU}$, and $\overrightarrow{UR}$ have the same direction.

Problem 4

Vector $\vec{p}$ has a length of 5 units and vector $\vec{q} = 3\vec{p}$. Determine the length of vector $\vec{q}$.

Solution: Given $|\vec{p}| = 5$ units and $\vec{q} = 3\vec{p}$.

To determine the length of vector $\vec{q}$, we use the property of scalar multiplication:

Therefore, the length of vector $\vec{q}$ is 15 units.

Problem 5

Given points $A(2, 3, -1)$, $B(5, -2, 4)$, and $C$ lies on the line passing through $A$ and $B$ such that $\overrightarrow{AC} = 2\overrightarrow{AB}$. Determine the coordinates of point $C$.

Solution: First, we determine the vector $\overrightarrow{AB}$:

Then, we use the relationship $\overrightarrow{AC} = 2\overrightarrow{AB}$ to determine the vector $\overrightarrow{AC}$:

Next, we determine the coordinates of point C:

Therefore, the coordinates of point C are $C(8, -7, 9)$.