Question

Find the following vectors. The vector that is 3 times \langle 3,-5\rangle plus -9 times \langle 6,0\rangle

Short answer

Answer: The resulting vector is <-45, -15>.

Step-by-step solution

Multiply the first vector by 3

To multiply the vector \langle 3, -5\rangle by 3, we multiply each of its components by 3: \[3\langle 3, -5\rangle = \langle 3(3), 3(-5) \rangle = \langle 9, -15 \rangle\]

Multiply the second vector by -9

To multiply the vector \langle 6, 0\rangle by -9, we multiply each of its components by -9: \[-9\langle 6, 0\rangle = \langle -9(6), -9(0) \rangle = \langle -54, 0 \rangle\]

Add the resulting vectors from steps 1 and 2

Now add the resulting vectors from steps 1 and 2 to obtain the final vector: \[\langle 9, -15\rangle + \langle -54, 0 \rangle = \langle 9 + (-54), -15 + 0 \rangle = \langle -45, -15 \rangle\] The vector that is 3 times \langle 3, -5\rangle plus -9 times \langle 6, 0\rangle is \(\boxed{\langle -45, -15\rangle}\).

Key concepts

Vector Multiplication

Vector multiplication involves multiplying each component of a vector by a scalar (a number). This process scales the vector, either increasing or decreasing its magnitude. When we multiply the vector \( \langle 3, -5\rangle \) by 3, each component is multiplied individually:

• Multiply the x-component: \( 3 \times 3 = 9 \)
• Multiply the y-component: \( 3 \times (-5) = -15 \)

This gives us the new vector \( \langle 9, -15\rangle \). Similarly, for the vector \( \langle 6, 0\rangle \) multiplied by \(-9\), we get:

• \(-9 \times 6 = -54\)
• \(-9 \times 0 = 0\)

Thus, the vector becomes \( \langle -54, 0\rangle \). Multiplying vectors by scalars alters their size but not their direction, unless the scalar is negative. Then, the vector points in the opposite direction.

Vector Addition

Adding vectors involves adding the corresponding components of the vectors together. This operation results in a new vector. In our initial exercise, after finding:

• \( \langle 9, -15\rangle \)
• \( \langle -54, 0\rangle \)

We add these together by summing their x-components and y-components separately:

• X-component: \( 9 + (-54) = -45 \)
• Y-component: \( -15 + 0 = -15 \)

Thus, the resulting vector is \( \langle -45, -15\rangle \). This combines the effects of both vectors, taking into account their directions and magnitudes, resulting in a vector that represents the cumulative effect of both.

Mathematical Problem Solving

Approaching mathematical problems involving vectors requires a clear understanding of vector operations like multiplication and addition. To solve such problems effectively:

• Start by identifying the vectors and operations required (like scalar multiplication).
• Perform operations step by step, focusing on one vector at a time.
• Be careful with signs, especially when multiplying by negative scalars, as these can change the vector's direction.
• After computing each step, check your calculations to ensure they are correct.
• Finally, combine your results, like through vector addition, to find your answer.

This process helps break down complicated problems into manageable steps, making it easier to understand and solve vector-related problems.