t) Let a , b , c and y be the three dimensional vectors

a =4j +2k ,b =−3i +4j +4k ,c =3i −j ,y =8i −7j
Perform the following operations on these vectors:
(a) c⋅a +a⋅y =
(b) (a ⋅b )a =
(c) ((c⋅c)a )⋅a =

I will try one. You try the rest.

c dot a = 3i − 1j + 0 k dot 0 i + 4j + 2k = 3*0-4*1 +0*2 = -4
a dot y = 0 i + 4j + 2k dot 8 i -7j + 0k = 0*8 -4*7 +2*0 = - 28
so -32

a=<0,4,2> , b=<-3,4,4> , c=<3,-1,0> and y=<8,-7,0>

I will do b)
you do the rest and let me know what you got

b) (a ⋅b )a
= (0 + 16 + 8)<0,4,2> , which is a scalar times a vector
= 24<0,4,2>
= <0,96,48>

LOL , well Carl better do the last one !

(a) c⋅a +a⋅y =

To calculate the dot product of two vectors, we multiply their corresponding components and sum them up. Here are the calculations:

c⋅a = (3i - j) ⋅ (4j + 2k)
= 0⋅4 + (-1)⋅2 + 0⋅0
= -2

a⋅y = (4j + 2k) ⋅ (8i - 7j)
= 0⋅8 + 4⋅0 + 2⋅(-7)
= -14

Now we can add the results together:

c⋅a + a⋅y = -2 + (-14)
= -16

So, c⋅a + a⋅y = -16.

(b) (a ⋅b )a =
To find this result, we need to calculate the dot product of vectors a and b first:

a⋅b = (4j + 2k) ⋅ (-3i + 4j + 4k)
= 0⋅(-3) + 4⋅4 + 2⋅4
= 16 + 8
= 24

Now we can multiply the dot product with vector a:

(a⋅b )a = 24(4j + 2k)
= 96j + 48k

So, (a ⋅b )a = 96j + 48k.

(c) ((c⋅c)a )⋅a =
First, we need to calculate the dot product of vector c with itself:

c⋅c = (3i - j) ⋅ (3i - j)
= 3⋅3 + (-1)⋅(-1) + 0⋅0
= 9 + 1
= 10

Now we can multiply the dot product with vector a:

((c⋅c)a )⋅a = 10(4j + 2k)
= 40j + 20k

So, ((c⋅c)a )⋅a = 40j + 20k.

To perform the given operations on the vectors, we need to understand the concepts of dot product (·) and scalar multiplication.

The dot product of two vectors is given by the formula:

a · b = |a| |b| cosθ

Where |a| and |b| are the magnitudes of vectors a and b, respectively, and θ is the angle between them.

Scalar multiplication involves multiplying a vector by a scalar (a real number).

Now, let's solve the given operations step-by-step:

(a) c⋅a + a⋅y:

To find c⋅a, we calculate the dot product of vectors c and a:

c⋅a = (3i - j) · (0i + 4j + 2k)
= (3)(0) + (-1)(4) + (0)(2)
= -4

Then, we find a⋅y by calculating the dot product of vectors a and y:

a⋅y = (0i + 4j + 2k) · (8i - 7j)
= (0)(8) + (4)(-7) + (2)(0)
= -28

Finally, we add these two results:

c⋅a + a⋅y = -4 + (-28)
= -32

Therefore, c⋅a + a⋅y is equal to -32.

(b) (a⋅b)a:

First, we calculate the dot product of vectors a and b:

a⋅b = (0i + 4j + 2k) · (-3i + 4j + 4k)
= (0)(-3) + (4)(4) + (2)(4)
= 16

Then, we multiply this result by vector a:

(a⋅b)a = 16(0i + 4j + 2k)
= 16(0i) + 16(4j) + 16(2k)
= 0i + 64j + 32k

Thus, (a⋅b)a is equal to 0i + 64j + 32k.

(c) ((c⋅c)a)⋅a:

First, we find c⋅c, the dot product of vector c with itself:

c⋅c = (3i - j) · (3i - j)
= (3)(3) + (-1)(-1) + (0)(0)
= 9 + 1 + 0
= 10

Next, we multiply this result by vector a:

((c⋅c)a) = 10(0i + 4j + 2k)
= 10(0i) + 10(4j) + 10(2k)
= 0i + 40j + 20k

Finally, we calculate the dot product of this resulting vector with vector a:

((c⋅c)a)⋅a = (0i + 40j + 20k) · (0i + 4j + 2k)
= (0)(0) + (40)(4) + (20)(2)
= 160 + 40
= 200

Thus, ((c⋅c)a)⋅a is equal to 200.

In summary, the answers to the given operations are:
(a) c⋅a + a⋅y = -32
(b) (a⋅b)a = 0i + 64j + 32k
(c) ((c⋅c)a)⋅a = 200