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If a = 1 \ 4 and b = -3 \ 4 , find 2a + b.

Mathematics

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4 parts
6

(−112)\begin{pmatrix} -1 \\ 12 \end{pmatrix}

7

4.5 km\text{4.5 km}

8

a + b

9

steep gradient\text{steep gradient}

Step-by-step solution

7 steps

Here are the solutions to the questions from the image:

Part 6· 2 steps

If a=(14)a = \begin{pmatrix} 1 \\ 4 \end{pmatrix} and b=(−34)b = \begin{pmatrix} -3 \\ 4 \end{pmatrix}, find 2a+b2a + b.

  1. Calculate $2a$.

    2a=2(14)=(2×12×4)=(28)2a = 2 \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \times 1 \\ 2 \times 4 \end{pmatrix} = \begin{pmatrix} 2 \\ 8 \end{pmatrix}

  2. Add 2a2a and bb.

    2a+b=(28)+(−34)=(2+(−3)8+4)=(−112)2a + b = \begin{pmatrix} 2 \\ 8 \end{pmatrix} + \begin{pmatrix} -3 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 + (-3) \\ 8 + 4 \end{pmatrix} = \begin{pmatrix} -1 \\ 12 \end{pmatrix}

    The vector 2a+b2a + b is (−112)\boxed{\begin{pmatrix} -1 \\ 12 \end{pmatrix}}.

Part 7

4.5 km\text{4.5 km}

If the scale is 1:10 000, what length will 45 cm on the map represent in km?

  1. Calculate the real distance in centimeters.

    A scale of 1:10 000 means 1 cm on the map represents 10 000 cm in real life.
    For 45 cm on the map, the real distance is:

    45 cm×10000=450000 cm45 \text{ cm} \times 10 000 = 450 000 \text{ cm}

  2. Convert the real distance from centimeters to meters.

    Since 1 m=100 cm1 \text{ m} = 100 \text{ cm}:

    450000 cm=450000100 m=4500 m450 000 \text{ cm} = \frac{450 000}{100} \text{ m} = 4500 \text{ m}

  3. Convert the real distance from meters to kilometers.

    Since 1 km=1000 m1 \text{ km} = 1000 \text{ m}:

    4500 m=45001000 km=4.5 km4500 \text{ m} = \frac{4500}{1000} \text{ km} = 4.5 \text{ km}

    The length represented is 4.5 km\boxed{\text{4.5 km}}.

Part 8a + b

Find vector x in terms of a and b.

  1. Apply the triangle law of vector addition.

    The diagram shows vector 'a' and vector 'b' connected head-to-tail. Vector 'x' is the resultant vector that connects the tail of 'a' to the head of 'b'.

  2. Write the vector equation.

    According to the triangle law of vector addition, the resultant vector 'x' is the sum of vectors 'a' and 'b'.

    x⃗=a⃗+b⃗\vec{x} = \vec{a} + \vec{b}

    Vector x in terms of a and b is a+b\boxed{a + b}.

Part 9

steep gradient\text{steep gradient}

Contour lines that are close together show that the land has a ____ quantities.

Contour lines on a map connect points of equal elevation. When these lines are drawn close to each other, it indicates a rapid change in elevation over a short horizontal distance.
Contour lines that are close together show that the land has a steep gradient\boxed{\text{steep gradient}}.

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