Question

Answer fast..!!!!
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Answer 1

the ans is

u⁶-2u⁵+2u⁴-4u³-8u

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Answer 2

Answer:

the ans is

u⁶-2u⁵+2u⁴-4u³-8u

Step-by-step explanation:

The Dot Product

In this section, we will now concentrate on the vector operation called the dot product. The dot

product of two vectors will produce a scalar instead of a vector as in the other operations that we

examined in the previous section. The dot product is equal to the sum of the product of the

horizontal components and the product of the vertical components.

If v = a1 i + b1 j and w = a2 i + b2 j are vectors then their dot product is given by:

v · w = a1 a2 + b1 b2

Properties of the Dot Product

If u, v, and w are vectors and c is a scalar then:

u · v = v · u

u · (v + w) = u · v + u · w

0 · v = 0

v · v = || v || 2

(cu) · v = c(u · v) = u · (cv)

Example 1: If v = 5i + 2j and w = 3i – 7j then find v · w.

Solution:

v · w = a1 a2 + b1 b2

v · w = (5)(3) + (2)(-7)

v · w = 15 – 14

v · w = 1

Example 2: If u = –i + 3j, v = 7i – 4j and w = 2i + j then find (3u) · (v + w).

Solution:

Find 3u

3u = 3(–i + 3j)

3u = –3i + 9j

Find v + w

v + w = (7i – 4j) + (2i + j)

v + w = (7 + 2) i + (–4 + 1) j

v + w = 9i – 3j

Example 2 (Continued):

Find the dot product between (3u) and (v + w)

(3u) · (v + w) = (–3i + 9j) · (9i – 3j)

(3u) · (v + w) = (–3)(9) + (9)(-3)

(3u) · (v + w) = –27 – 27

(3u) · (v + w) = –54

An alternate formula for the dot product is available by using the angle between the two vectors.

If v and w are two nonzero vectors and θ is the smallest nonnegative angle between them then

their dot product is given:

v · w = || v || || w || cos θ

This same equation could be solved for theta if the angle between the vectors needed to be

determined.

1 cos

v w

v w

θ − ⋅ =

Example 3: If u = 6i – 2j and v = 3i + 5j then find the angle θ between the vectors. Round the

answer to the nearest tenth of a degree, if necessary.

Solution:

Find the magnitude of u

2 2 u ab = +

2 2

u = +− (6) ( 2)

u = + 36 4

u = 40

u = 2 10

Find the magnitude of v

2 2 v ab = +

2 2

v = + (3) (5)

v = +9 25

v = 34