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For the following vectors, (a) find the dot product v•w ; (b) find the angle between v and w , (c) state whether the vectors are parallel, octagonal, or neither. V=-3i-4j, w=6i+8j

A- v•w
B-the angle between v and w is theta ^•?
C- the vectors v and w are?

1 Answer

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A) v • w = (-3)(6) + (-4)(8) = -18 - 32 = -50
B) The angle between vectors v and w can be found using the formula: cos(theta) = (v • w) / (||v|| ||w||), where ||v|| and ||w|| are the magnitudes of vectors v and w respectively.

First, we need to find ||v|| and ||w||:

||v|| = sqrt((-3)^2 + (-4)^2) = 5
||w|| = sqrt((6)^2 + (8)^2) = 10

Now, we can substitute in the values to get:

cos(theta) = (-50) / (5 * 10) = -1
theta = arccos(-1) = pi radians or 180 degrees.

Therefore, the angle between vectors v and w is 180 degrees.

C) Two vectors are parallel if their directions are the same, which can be determined by comparing their unit vectors.

The unit vector of v is:

v_hat = v / ||v|| = (-3/5)i + (-4/5)j

The unit vector of w is:

w_hat = w / ||w|| = (6/10)i + (8/10)j = (3/5)i + (4/5)j

We can see that the unit vectors are in opposite directions, which means that the vectors are anti-parallel or opposite. Therefore, the vectors v and w are neither parallel nor orthogonal.
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