Question

Determine whether the vectors u and v are parallel, orthogonal, or neither.

Answer 1

Answer:

Orthogonal.

Step-by-step explanation:

Given:

u = <10, 0>

v = <0, -9>

In unit vector notation, the above vectors can be re-written as:

u = 10i + 0j

v = 0i - 9j

Now, note the following:

(i) two vectors, u and v, are parallel to each other if one is a scalar multiple of the other. i.e

u = kv

or

v = ku

for some nonzero value of a scalar k.

(ii) two vectors are orthogonal if their dot product gives zero. i.e

u . v = 0

Let's use the explanations above to determine whether the given vectors are parallel or orthogonal.

(a) If parallel

u = k v

10i + 0j = k (0i - 9j)   ?

When k = 1, the above equation becomes

10i + 0j  ≠  0i - 9j

When k = 2,

10i + 0j ≠ 2(0i - 9j)

10i + 0j ≠ 0i - 18j

Since we cannot find any value of k for which u = kv or v = ku, then the two vectors are not parallel to each other.

(b) If Orthogonal

u.v = (10i + 0j) . (0i - 9j)  

[multiply the i components together, and add the result to the multiplication of the j components]

u.v = (10i * 0i) + (0j * 9j)

u.v = (0) + (0)

u.v = 0

Since the dot product of the two vectors gave zero, then the two vectors are orthogonal.