# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/vector-operations/scalar-multiplication Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/vector-operations/scalar-multiplication/en.mdx Learn scalar multiplication of vectors: scale magnitude, change direction with negative values. Learn properties, calculations, and physics applications. --- ## 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. Visible text: Scalar multiplication of a vector is an operation involving multiplication between a real number (scalar) and a vector . 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. Visible text: If is a real number (scalar) and is a vector, then the scalar multiplication of a vector is denoted as 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 $$|k|\text{ times}$$ the magnitude of the original vector. Visible text: 1. If (positive), then the resulting vector has the same direction as the original vector. 2. If (negative), then the resulting vector has a direction opposite to the original vector. 3. If , then the resulting vector is a zero vector. 4. The magnitude (length) of the resulting vector is 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 $$|k|\text{ times}$$. The direction of the vector depends on the sign of $$k$$: Visible text: Geometrically, scalar multiplication of a vector changes the length (magnitude) of the vector by . The direction of the vector depends on the sign of : - If $$k > 0$$, the direction of the vector remains unchanged - If $$k < 0$$, the direction of the vector is opposite to the original vector $$|k \cdot \vec{v}| = |k| \cdot |\vec{v}|$$ Visible text: - If , the direction of the vector remains unchanged - If , the direction of the vector is opposite to the original vector ### Algebraically If $$\vec{v} = (v_1, v_2, v_3)$$ is a vector in $$3$$-dimensional space, then: Visible text: If is a vector in -dimensional space, then: Component: MathContainer Children: ```math k \cdot \vec{v} = k(v_1, v_2, v_3) = (k \cdot v_1, k \cdot v_2, k \cdot v_3) ``` In unit vector notation: Component: MathContainer Children: ```math k \cdot \vec{v} = k(v_1\vec{i} + v_2\vec{j} + v_3\vec{k}) = k \cdot v_1\vec{i} + k \cdot v_2\vec{j} + k \cdot v_3\vec{k} ``` ## Examples of Scalar Multiplication of Vectors ### First Example Given the vector $$\vec{a} = 2\vec{i} + 3\vec{j} + 5\vec{k}$$. Determine the result of multiplication $$2\vec{a}$$. Visible text: Given the vector . Determine the result of multiplication . **Solution:** Component: MathContainer Children: ```math 2\vec{a} = 2(2\vec{i} + 3\vec{j} + 5\vec{k}) ``` ```math = 4\vec{i} + 6\vec{j} + 10\vec{k} ``` ### Second Example Given the vector $$\vec{v} = (4, -2, 3)$$. Determine the result of $$-3\vec{v}$$. Visible text: Given the vector . Determine the result of . **Solution:** Component: MathContainer Children: ```math -3\vec{v} = -3(4, -2, 3) ``` ```math = (-12, 6, -9) ``` 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. Visible text: 1. **Force and Acceleration**: If an object with mass experiences acceleration , then the force acting on the object is . 2. **Velocity**: If an object moves with velocity for a time , then the displacement of the object is . 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 $$5 \text{ units}$$ 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 $$C$$. Visible text: 1. Given the vector . Determine the result of . 2. Vectors and . Determine the vector . 3. Given the vector . If and , prove that all three vectors have the same direction. 4. Vector has a length of and vector . Determine the length of vector . 5. Given points , , and lies on the line passing through and such that . Determine the coordinates of point . ## Answer Key ### First Problem Given the vector $$\vec{a} = 3\vec{i} - 4\vec{j} + 2\vec{k}$$. Determine the result of $$-2\vec{a}$$. Visible text: Given the vector . Determine the result of . **Solution:** Component: MathContainer Children: ```math -2\vec{a} = -2(3\vec{i} - 4\vec{j} + 2\vec{k}) ``` ```math = -6\vec{i} + 8\vec{j} - 4\vec{k} ``` Therefore, the result of $$-2\vec{a}$$ is $$-6\vec{i} + 8\vec{j} - 4\vec{k}$$. Visible text: Therefore, the result of is . ### Second Problem Vectors $$\vec{u} = (2, -5, 1)$$ and $$\vec{v} = (-4, 3, 6)$$. Determine the vector $$2\vec{u} - 3\vec{v}$$. Visible text: Vectors and . Determine the vector . **Solution:** Component: MathContainer Children: ```math 2\vec{u} = 2(2, -5, 1) = (4, -10, 2) ``` ```math 3\vec{v} = 3(-4, 3, 6) = (-12, 9, 18) ``` ```math 2\vec{u} - 3\vec{v} = (4, -10, 2) - (-12, 9, 18) ``` ```math = (4 - (-12), -10 - 9, 2 - 18) ``` ```math = (4 + 12, -10 - 9, 2 - 18) ``` ```math = (16, -19, -16) ``` Therefore, the vector $$2\vec{u} - 3\vec{v}$$ is $$(16, -19, -16)$$ or $$16\vec{i} - 19\vec{j} - 16\vec{k}$$. Visible text: Therefore, the vector is or . ### Third Problem 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. Visible text: Given the vector . If and , 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: - $$\overrightarrow{BU} = 0.65 \cdot \overrightarrow{BR}$$ - $$\overrightarrow{UR} = 0.35 \cdot \overrightarrow{BR}$$ Visible text: - - Let's check if $$\overrightarrow{BU} + \overrightarrow{UR} = \overrightarrow{BR}$$: Visible text: Let's check if : Component: MathContainer Children: ```math \overrightarrow{BU} + \overrightarrow{UR} = 0.65 \cdot \overrightarrow{BR} + 0.35 \cdot \overrightarrow{BR} ``` ```math = (0.65 + 0.35) \cdot \overrightarrow{BR} ``` ```math = 1 \cdot \overrightarrow{BR} ``` ```math = \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 $$R$$. Visible text: This result shows that , which aligns with the vector addition law for collinear points , , and . 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 $$\overrightarrow{BR}$$. Visible text: Since and , where the scalar factors are positive ( and ), all three vectors have the same direction. Positive scalar factors mean that these vectors point in the same direction as the reference vector . Therefore, it is proven that the three vectors $$\overrightarrow{BR}$$, $$\overrightarrow{BU}$$, and $$\overrightarrow{UR}$$ have the same direction. Visible text: Therefore, it is proven that the three vectors , , and have the same direction. ### Fourth Problem Vector $$\vec{p}$$ has a length of $$5 \text{ units}$$ and vector $$\vec{q} = 3\vec{p}$$. Determine the length of vector $$\vec{q}$$. Visible text: Vector has a length of and vector . Determine the length of vector . **Solution:** Given $$|\vec{p}| = 5 \text{ units}$$ and $$\vec{q} = 3\vec{p}$$. Visible text: **Solution:** Given and . To determine the length of vector $$\vec{q}$$, we use the property of scalar multiplication: Visible text: To determine the length of vector , we use the property of scalar multiplication: Component: MathContainer Children: ```math |\vec{q}| = |3\vec{p}| ``` ```math = |3| \cdot |\vec{p}| ``` ```math = 3 \cdot 5 ``` ```math = 15 ``` Therefore, the length of vector $$\vec{q}$$ is $$15 \text{ units}$$. Visible text: Therefore, the length of vector is . ### Fifth Problem 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 $$C$$. Visible text: Given points , , and lies on the line passing through and such that . Determine the coordinates of point . **Solution:** First, we determine the vector $$\overrightarrow{AB}$$: Visible text: **Solution:** First, we determine the vector : Component: MathContainer Children: ```math \overrightarrow{AB} = B - A ``` ```math = (5, -2, 4) - (2, 3, -1) ``` ```math = (5-2, -2-3, 4-(-1)) ``` ```math = (3, -5, 5) ``` Then, we use the relationship $$\overrightarrow{AC} = 2\overrightarrow{AB}$$ to determine the vector $$\overrightarrow{AC}$$: Visible text: Then, we use the relationship to determine the vector : Component: MathContainer Children: ```math \overrightarrow{AC} = 2\overrightarrow{AB} ``` ```math = 2(3, -5, 5) ``` ```math = (6, -10, 10) ``` Next, we determine the coordinates of point $$C$$: Visible text: Next, we determine the coordinates of point : Component: MathContainer Children: ```math \overrightarrow{AC} = C - A ``` ```math (6, -10, 10) = C - (2, 3, -1) ``` ```math C = (6, -10, 10) + (2, 3, -1) ``` ```math C = (6+2, -10+3, 10+(-1)) ``` ```math C = (8, -7, 9) ``` Therefore, the coordinates of point $$C$$ are $$C(8, -7, 9)$$. Visible text: Therefore, the coordinates of point are .