Question

Suppose that →p,→q and →r are three non-coplanar vectors in R3. Let the components of a vector →s along →p,→q, and →r be 4,3, and 5, respectively. If the components of this vector →s along (−→p+→q+→r), (→p−→q+→r), and (−→p−→q+→r), are x,y, and z, respectively, then the value of 2x+y+z is

Answer 1

To solve for x, y, and z, we used scalar components and projection of vectors on the new base vectors created by linear combinations of \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\). The scalar components of the vector s along the new vectors were found, and then we computed 2x + y + z using these components.

Step-by-step explanation:

To find the values of x, y, and z for the vector \(\vec{s}\) projected along the new vectors, we will use the concept of scalar components and projection of vectors. Since the vector \(\vec{s}\) has components 4, 3, and 5 along \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) respectively, and we are given three new base vectors which are linear combinations of \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\), we can find the projections using vector addition.

The new vectors and their scalar components relative to \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) are as follows:

• New vector 1: \(\vec{s}_1 = -\vec{p} + \vec{q} + \vec{r}\), so its components along \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) are -1, 1, and 1, respectively. This means x = -4 + 3 + 5.
• New vector 2: \(\vec{s}_2 = \vec{p} - \vec{q} + \vec{r}\), so y = 4 - 3 + 5.
• New vector 3: \(\vec{s}_3 = -\vec{p} - \vec{q} + \vec{r}\), so z = -4 - 3 + 5.

The sum to find 2x + y + z can now be computed by substituting the values calculated above.