Question

Given vectors u and v, find (a) 5u (b) 5u+4v (c) v-4u.

u=2i, v=3i+ 8j
(a) 5u =
(b) 5u+4v=
(c) v-4u =
(Type your answer in terms of i and j.)

Answer 1

The operations on vectors u and v yield: (a) 5u = 10i, (b) 5u + 4v = 22i + 32j, and (c) v - 4u = -5i + 8j.

Step-by-step explanation:

To find the operations on the given vectors u and v, we perform scalar multiplication and vector addition:

• (a) 5u: To find 5 times the vector u, we simply multiply each component of u by 5.
• Since u = 2i, we have 5u = 5(2i) = 10i.
• (b) 5u + 4v: First, calculate 5u as done in part (a). Then multiply each component of v by 4.
• Since v = 3i + 8j, we have 4v = 4(3i) + 4(8j) = 12i + 32j.
• Now add 5u and 4v: 5u + 4v = 10i + (12i + 32j) = (10i + 12i) + 32j = 22i + 32j.
• (c) v - 4u: Multiply each component of u by 4 and subtract from v. Since 4u = 4(2i) = 8i,
• we have v - 4u = (3i + 8j) - 8i = (3i - 8i) + 8j = -5i + 8j.

Answer 2

(a) 5u = 10i

(b) 5u +4v = 22i +32j

(c) v -4u = -5i +8j

Step-by-step explanation:

Given vectors u = 2i and v = 3i +8j, you want the values of ...

• 5u
• 5u +4v
• v -4u

Vector sum

The product of a scalar and a vector is the vector comprised of the products of that scalar and each of the vector components. The multiplication by a scalar distributes over the vector components.

The sum of two vectors is the vector comprised of the sums of corresponding components. Effectively the unit vectors can be treated as though they were variables: like terms can be combined.

(a) 5u

5u = 5(2i) = 10i

(b) 5u +4v

5u +4v = 5(2i) +4(3i +8j) = 10i +12i +32j = 22i +32j

(c) v -4u

v -4u = (3i +8j) -4(2i) = 3i -8i +8j = -5i +8j

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Additional comment

Many graphing and scientific calculators can perform vector operations for you.

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