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Vectors and Transformations - Vector Addition, Subtraction, and Scalar Multiplication

Grade 10IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition of a Vector: A quantity that has both magnitude (size) and direction.

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Column Vectors: Represented as (xy)\begin{pmatrix} x \\ y \end{pmatrix}, where xx is the horizontal displacement and yy is the vertical displacement.

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Vector Addition: Geometrically, this follows the 'nose-to-tail' rule. Algebraically, it involves adding the corresponding xx and yy components.

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Vector Subtraction: Subtracting a vector is the same as adding its opposite. Geometrically, A⃗−B⃗\vec{A} - \vec{B} is the vector from the tip of BB to the tip of AA.

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Scalar Multiplication: Multiplying a vector by a real number kk changes its magnitude. If kk is negative, it reverses the direction.

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Parallel Vectors: Two vectors are parallel if one is a scalar multiple of the other (e.g., a=kb\mathbf{a} = k\mathbf{b}).

📐Formulae

Addition: (x1y1)+(x2y2)=(x1+x2y1+y2)\text{Addition: } \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}

Subtraction: (x1y1)−(x2y2)=(x1−x2y1−y2)\text{Subtraction: } \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} - \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}

Scalar Multiplication: k(xy)=(kxky)\text{Scalar Multiplication: } k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}

Magnitude of v=(xy):∣v∣=x2+y2\text{Magnitude of } \mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}: |\mathbf{v}| = \sqrt{x^2 + y^2}

💡Examples

Problem 1:

Given a=(4−2)\mathbf{a} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} and b=(−15)\mathbf{b} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}, calculate 3a+b3\mathbf{a} + \mathbf{b}.

Solution:

3(4−2)+(−15)=(12−6)+(−15)=(11−1)3 \begin{pmatrix} 4 \\ -2 \end{pmatrix} + \begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} 12 \\ -6 \end{pmatrix} + \begin{pmatrix} -1 \\ 5 \end{pmatrix} = \begin{pmatrix} 11 \\ -1 \end{pmatrix}

Explanation:

First, perform scalar multiplication by multiplying each component of vector a\mathbf{a} by 3. Then, add the resulting xx-components (12+(−1)12 + (-1)) and yy-components (−6+5-6 + 5).

Problem 2:

Are the vectors u=(23)\mathbf{u} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} and v=(1015)\mathbf{v} = \begin{pmatrix} 10 \\ 15 \end{pmatrix} parallel?

Solution:

Yes, they are parallel because v=5u\mathbf{v} = 5\mathbf{u}.

Explanation:

To check if vectors are parallel, determine if one is a scalar multiple of the other. Since 10=5×210 = 5 \times 2 and 15=5×315 = 5 \times 3, vector v\mathbf{v} is exactly 5 times vector u\mathbf{u}, confirming they share the same direction.

Problem 3:

Find the magnitude of the vector PQ⃗\vec{PQ} where P=(1,2)P = (1, 2) and Q=(4,6)Q = (4, 6).

Solution:

PQ⃗=(4−16−2)=(34)\vec{PQ} = \begin{pmatrix} 4-1 \\ 6-2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}. ∣PQ⃗∣=32+42=9+16=25=5|\vec{PQ}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.

Explanation:

First, find the column vector by subtracting the coordinates of the starting point from the end point. Then, use the Pythagorean theorem formula for magnitude.

Vector Addition, Subtraction, and Scalar Multiplication Revision - Grade 10 Maths IGCSE